6655:
11045:
6071:
2057:
1385:
43:
10799:
12081:
6650:{\displaystyle {\begin{aligned}p&={\frac {8c-3b^{2}}{8}}={\frac {8a_{2}a_{4}-3{a_{3}}^{2}}{8{a_{4}}^{2}}}\\q&={\frac {b^{3}-4bc+8d}{8}}={\frac {{a_{3}}^{3}-4a_{2}a_{3}a_{4}+8a_{1}{a_{4}}^{2}}{8{a_{4}}^{3}}}\\r&={\frac {-3b^{4}+256e-64bd+16b^{2}c}{256}}={\frac {-3{a_{3}}^{4}+256a_{0}{a_{4}}^{3}-64a_{1}a_{3}{a_{4}}^{2}+16a_{2}{a_{3}}^{2}a_{4}}{256{a_{4}}^{4}}}.\end{aligned}}}
2917:
2530:
885:
11040:{\displaystyle \left\{{\begin{array}{l}r_{1}={\frac {{\sqrt {\alpha }}+{\sqrt {\beta }}+{\sqrt {\gamma }}}{2}}\\r_{2}={\frac {{\sqrt {\alpha }}-{\sqrt {\beta }}-{\sqrt {\gamma }}}{2}}\\r_{3}={\frac {-{\sqrt {\alpha }}+{\sqrt {\beta }}-{\sqrt {\gamma }}}{2}}\\r_{4}={\frac {-{\sqrt {\alpha }}-{\sqrt {\beta }}+{\sqrt {\gamma }}}{2}}{\text{.}}\end{array}}\right.}
10398:
7509:, and thus that the depressed equation is bi-quadratic, and may be solved by an easier method (see above). This was not a problem at the time of Ferrari, when one solved only explicitly given equations with numeric coefficients. For a general formula that is always true, one thus needs to choose a root of the cubic equation such that
12739:
11701:
10788:
7908:
9207:
5361:
12487:, we obtain expression for the roots. In fact we obtain, apparently, several expressions, depending on the numbering of the roots of the cubic polynomial and of the signs given to their square roots. All these different expressions may be deduced from one of them by simply changing the numbering of the
2720:
8557:
1380:{\displaystyle {\begin{aligned}\Delta ={}&256a^{3}e^{3}-192a^{2}bde^{2}-128a^{2}c^{2}e^{2}+144a^{2}cd^{2}e-27a^{2}d^{4}\\&+144ab^{2}ce^{2}-6ab^{2}d^{2}e-80abc^{2}de+18abcd^{3}+16ac^{4}e\\&-4ac^{3}d^{2}-27b^{4}e^{2}+18b^{3}cde-4b^{3}d^{3}-4b^{2}c^{3}e+b^{2}c^{2}d^{2}\end{aligned}}}
8251:
2296:
10098:
3122:
8048:
2707:
3451:
12523:
12076:{\displaystyle {\begin{aligned}s_{0}&={\tfrac {1}{2}}(x_{0}+x_{1}+x_{2}+x_{3}),\\s_{1}&={\tfrac {1}{2}}(x_{0}-x_{1}+x_{2}-x_{3}),\\s_{2}&={\tfrac {1}{2}}(x_{0}+x_{1}-x_{2}-x_{3}),\\s_{3}&={\tfrac {1}{2}}(x_{0}-x_{1}-x_{2}+x_{3}),\end{aligned}}}
13186:
7670:
10590:
766:
Moreover, the area of the region between the secant line and the quartic below the secant line equals the area of the region between the secant line and the quartic above the secant line. One of those regions is disjointed into sub-regions of equal area.
7730:
761:
8870:
7093:
3535:
9489:
5727:
by the following simple change of variable. All formulas are simpler and some methods work only in this case. The roots of the original quartic are easily recovered from that of the depressed quartic by the reverse change of variable.
5181:
3229:, any of the three cube roots in the complex plane can be used, although if one of them is real that is the natural and simplest one to choose. The mathematical expressions of these last four terms are very similar to those of their
2912:{\displaystyle {\begin{aligned}S&={\frac {1}{2}}{\sqrt {-{\frac {2}{3}}\ p+{\frac {1}{3a}}\left(Q+{\frac {\Delta _{0}}{Q}}\right)}}\\Q&={\sqrt{\frac {\Delta _{1}+{\sqrt {\Delta _{1}^{2}-4\Delta _{0}^{3}}}}{2}}}\end{aligned}}}
8683:
8851:
3682:
6917:
8290:
7262:
2525:{\displaystyle {\begin{aligned}x_{1,2}\ &=-{\frac {b}{4a}}-S\pm {\frac {1}{2}}{\sqrt {-4S^{2}-2p+{\frac {q}{S}}}}\\x_{3,4}\ &=-{\frac {b}{4a}}+S\pm {\frac {1}{2}}{\sqrt {-4S^{2}-2p-{\frac {q}{S}}}}\end{aligned}}}
9506:
The above solution shows that a quartic polynomial with rational coefficients and a zero coefficient on the cubic term is factorable into quadratics with rational coefficients if and only if either the resolvent cubic
8091:
12288:
7490:
5512:
4841:
11706:
9494:
The symmetries in this solution are as follows. There are three roots of the cubic, corresponding to the three ways that a quartic can be factored into two quadratics, and choosing positive or negative values of
4480:
4690:
1769:
568:, in which the lengths of two crossed ladders, each based against one wall and leaning against another, are given along with the height at which they cross, and the distance between the walls is to be found.
11561:
5850:
583:
3199:
10393:{\displaystyle \left\{{\begin{array}{l}r_{1}+r_{2}+r_{3}+r_{4}=0\\(r_{1}+r_{2})(r_{3}+r_{4})=-\alpha \\(r_{1}+r_{3})(r_{2}+r_{4})=-\beta \\(r_{1}+r_{4})(r_{2}+r_{3})=-\gamma {\text{.}}\end{array}}\right.}
12884:(pairs of lines) that pass through these points (this corresponds to the resolvent cubic, the pairs of lines being the Lagrange resolvents), and then use these linear equations to solve the quadratic.
12850:
2949:
11455:, which are the numbers obtained if one of the square roots is replaced by the symmetric one (or, what amounts to the same thing, if each of the three square roots is replaced by the symmetric one).
7923:
2552:
7372:
3944:
13019:
12528:
12441:
8875:
8295:
6076:
5186:
2954:
2725:
2557:
2301:
890:
3352:
13359: matrix: reducible quadratics correspond to this matrix being singular, which is equivalent to its determinant being zero, and the determinant is a homogeneous degree three polynomial in
12734:{\displaystyle {\begin{aligned}F_{1}(x)&=x^{2}+sx+{\frac {c}{2}}+{\frac {s^{2}}{2}}-{\frac {d}{2s}}\\F_{2}(x)&=x^{2}-sx+{\frac {c}{2}}+{\frac {s^{2}}{2}}+{\frac {d}{2s}}\end{aligned}}}
9321:
5046:
201:
2265:
2046:
318:
10783:{\displaystyle \left\{{\begin{array}{l}r_{1}+r_{2}+r_{3}+r_{4}=0\\r_{1}+r_{2}={\sqrt {\alpha }}\\r_{1}+r_{3}={\sqrt {\beta }}\\r_{1}+r_{4}={\sqrt {\gamma }}{\text{;}}\end{array}}\right.}
1638:
864:
13261:
13014:
2138:
7551:
3747:
1542:
428:
6809:
4329:
4265:
4171:
4077:
4041:
3603:
3567:
3270:
7903:{\displaystyle \left(y^{2}+{\frac {p}{2}}+m+{\sqrt {2m}}y-{\frac {q}{2{\sqrt {2m}}}}\right)\left(y^{2}+{\frac {p}{2}}+m-{\sqrt {2m}}y+{\frac {q}{2{\sqrt {2m}}}}\right)=0.}
3807:
3777:
1439:
9202:{\displaystyle {\begin{aligned}u^{2}(p+u^{2})^{2}-q^{2}&=u^{2}(t+v)^{2}-u^{2}(t-v)^{2}\\&=u^{2}\\&=u^{2}(2t)(2v)\\&=4u^{2}tv\\&=4u^{2}r\end{aligned}}}
12334:, these coefficients may be expressed as polynomials in the coefficients of the monic quartic. If, for simplification, we suppose that the quartic is depressed, that is
4293:
4005:
683:
475:
is credited with the discovery of the solution to the quartic in 1540, but since this solution, like all algebraic solutions of the quartic, requires the solution of a
6956:
3884:
3711:
3462:
3219:
9364:
5356:{\displaystyle {\begin{aligned}x_{1}&=+{\sqrt {z_{+}}},\\x_{2}&=-{\sqrt {z_{+}}},\\x_{3}&=+{\sqrt {z_{-}}},\\x_{4}&=-{\sqrt {z_{-}}}.\end{aligned}}}
4202:
4108:
3838:
3333:
3968:
3858:
3310:
3290:
12880:
There is an alternative solution using algebraic geometry In brief, one interprets the roots as the intersection of two quadratic curves, then finds the three
8572:
449:
is negative, it decreases to negative infinity and has a global maximum. In both cases it may or may not have another local maximum and another local minimum.
8752:
3608:
13222:
This pencil contains three reducible quadratics, each corresponding to a pair of lines, each passing through two of the four points, which can be done
3946:
The result is then correct, but misleading because it hides the fact that no cube root is needed in this case. In fact this case may occur only if the
8552:{\displaystyle {\begin{aligned}x^{4}+bx^{3}+cx^{2}+dx+e&=(x^{2}+sx+t)(x^{2}+ux+v)\\&=x^{4}+(s+u)x^{3}+(t+v+su)x^{2}+(sv+tu)x+tv\end{aligned}}}
6824:
89:
axis, there would be no real root (and four complex roots). The same reasoning applies in reverse to polynomial with a negative quartic coefficient.
8246:{\displaystyle x=-{a_{3} \over 4a_{4}}+{\pm _{1}{\sqrt {2m}}\pm _{2}{\sqrt {-\left(2p+2m\pm _{1}{{\sqrt {2}}q \over {\sqrt {m}}}\right)}} \over 2}.}
7148:
2140:
written out in full. This formula is too unwieldy for general use; hence other methods, or simpler formulas for special cases, are generally used.
479:
to be found, it could not be published immediately. The solution of the quartic was published together with that of the cubic by
Ferrari's mentor
433:
Since a quartic function is defined by a polynomial of even degree, it has the same infinite limit when the argument goes to positive or negative
12159:
7420:
6818:. The depressed equation may be rewritten (this is easily verified by expanding the square and regrouping all terms in the left-hand side) as
5377:
4701:
9340:
12331:
4355:
4563:
1649:
561:-axis must be found where it is tangent to a fixed line, and this requires the solution of a general quartic equation to be calculated.
577:
Given a light source and a spherical mirror, find the point on the mirror where the light will be reflected to the eye of an observer.
11523:
5737:
3133:
3117:{\displaystyle {\begin{aligned}\Delta _{0}&=c^{2}-3bd+12ae\\\Delta _{1}&=2c^{3}-9bcd+27b^{2}e+27ad^{2}-72ace\end{aligned}}}
11415:, then the choice of the square roots was a good one (again, by Vieta's formulas); otherwise, the roots of the polynomial will be
8284:
Descartes introduced in 1637 the method of finding the roots of a quartic polynomial by factoring it into two quadratic ones. Let
3339:
of the cubic function extended to the present context of the quartic. One may prefer to express it in a purely real way, by using
13603:
13484:
3292:
is a non-real complex number. In this case, either all roots are non-real or they are all real. In the latter case, the value of
12750:
9343:. This resolvent cubic is equivalent to the resolvent cubic given above (equation (1a)), as can be seen by substituting U = 2m.
492:
The proof that four is the highest degree of a general polynomial for which such solutions can be found was first given in the
8043:{\displaystyle y={\pm _{1}{\sqrt {2m}}\pm _{2}{\sqrt {-\left(2p+2m\pm _{1}{{\sqrt {2}}q \over {\sqrt {m}}}\right)}} \over 2},}
2702:{\displaystyle {\begin{aligned}p&={\frac {8ac-3b^{2}}{8a^{2}}}\\q&={\frac {b^{3}-4abc+8a^{2}d}{8a^{3}}}\end{aligned}}}
14147:
13971:
13707:
13697:
357:
of a square (or, equivalently, to the function defined by a quartic polynomial without terms of odd degree), having the form
3446:{\displaystyle S={\frac {1}{2}}{\sqrt {-{\frac {2}{3}}\ p+{\frac {2}{3a}}{\sqrt {\Delta _{0}}}\cos {\frac {\varphi }{3}}}}}
7275:
3889:
12349:
4905:(that is, if coefficients are restricted to be rational numbers) then there is an algorithm to determine whether or not
14327:
11154:
In order to determine the right sign of the square roots, one simply chooses some square root for each of the numbers
11089: such choices, because the consequence of replacing one of the square roots by the symmetric one is that the set
9233:
4963:
13930:
13845:
13518:
619:
111:
5518:
13963:
13922:
13181:{\displaystyle {\begin{aligned}F_{1}(X,Y,Z)&:=Y^{2}+pYZ+qXZ+rZ^{2},\\F_{2}(X,Y,Z)&:=YZ-X^{2}\end{aligned}}}
9969:. It also follows from Vieta's formulas, together with the fact that we are working with a depressed quartic, that
2186:
7665:{\displaystyle \left(y^{2}+{\frac {p}{2}}+m\right)^{2}=\left(y{\sqrt {2m}}-{\frac {q}{2{\sqrt {2m}}}}\right)^{2}.}
1986:
237:
13912:
1576:
786:
555:
cutter. To calculate its location relative to a triangulated surface, the position of a horizontal torus on the
13225:
13219:
are not both zero, and multiplying a quadratic form by a constant does not change its quadratic curve of zeros.
2063:
4869:
3716:
496:
in 1824, proving that all attempts at solving the higher order polynomials would be futile. The notes left by
14388:
13996:
13994:
Faucette, William M. (1996), "A Geometric
Interpretation of the Solution of the General Quartic Polynomial",
13725:
13662:
13621:
MacKay, R. J.; Oldford, R. W. (August 2000), "Scientific Method, Statistical Method and the Speed of Light",
34:
13546:
81:
axis, there would only be two real roots (and two complex roots). If all three local extrema were above the
14322:
14306:
1477:
548:
537:
485:
363:
47:
9352:
is a square root of a non-zero root of this resolvent (such a non-zero root exists except for the quartic
6750:
6733:
is a root of the original quartic and every root of the original quartic can be obtained by this process.
231:, or equation of the fourth degree, is an equation that equates a quartic polynomial to zero, of the form
14378:
14342:
2287:
4298:
4207:
441:
is positive, then the function increases to positive infinity at both ends; and thus the function has a
14187:
4113:
1828:
14140:
13427:
4331:
the above expression for the roots is correct but misleading, hiding the fact that the polynomial is
598:
13869:
13598:
7399:
4046:
4010:
3572:
544:. Here are examples of other geometric problems whose solution involves solving a quartic equation.
14332:
13608:
13489:
12087:
10808:
3546:
3246:
493:
461:
20:
3782:
3752:
3230:
1951:
There are some cases that do not seem to be covered, but in fact they cannot occur. For example,
1396:
756:{\displaystyle {\frac {FG}{GH}}={\frac {1+{\sqrt {5}}}{2}}=\varphi \;({\text{the golden ratio}}).}
14271:
13479:
11050:
Since, in general, there are two choices for each square root, it might look as if this provides
7088:{\displaystyle \left(y^{2}+{\frac {p}{2}}+m\right)^{2}=2my^{2}-qy+m^{2}+mp+{\frac {p^{2}}{4}}-r,}
3340:
565:
3530:{\displaystyle \varphi =\arccos \left({\frac {\Delta _{1}}{2{\sqrt {\Delta _{0}^{3}}}}}\right).}
14301:
14296:
14291:
14169:
13423:
9484:{\displaystyle \left\{{\begin{array}{l}s=-u\\2t=p+u^{2}+q/u\\2v=p+u^{2}-q/u\end{array}}\right.}
4486:
4332:
4272:
3984:
525:
215:
102:
12951:
3863:
3687:
14281:
14261:
12465:, and so the corresponding equation is solvable by the method described in the article about
8563:
3204:
609:
533:
10793:
the sign of the square roots will be dealt with below. The only solution of this system is:
4916:
is reducible and, if it is, how to express it as a product of polynomials of smaller degree.
4892:(that is, if coefficients are restricted to be real numbers) (or, more generally, over some
613:
14347:
14266:
14133:
14120:
14075:
14044:
14025:
13877:
13644:
13568:
12318:
9552:
root of the resolvent cubic, Euler's method uses all of them. Consider a depressed quartic
4180:
4086:
594:
572:
520:
is a solution of a quartic equation. The same is true for the intersection of a line and a
54:
13981:
13940:
13855:
4857:
are the roots of the factors, which may be computed using the formulas for the roots of a
3814:
8:
14160:
13958:
13594:
13475:
11647:
9966:
8678:{\displaystyle \left\{{\begin{array}{l}b=s+u\\c=t+v+su\\d=sv+tu\\e=tv\end{array}}\right.}
7123:
4543:
3336:
605:
27:
3315:
2274:≠ 0 are given in the following formula, which is deduced from the one in the section on
14383:
14337:
14204:
14199:
14125:
14092:
14061:
14013:
13894:
13837:
13776:
13742:
13679:
13507:
13386:
9533:
7135:
5641:
5084:
4858:
4177:
of the quartic by its second derivative, which is a linear polynomial. The simple root
4174:
3953:
3843:
3295:
3275:
2283:
354:
13812:
13795:
13339:
The reducible quadratics, in turn, may be determined by expressing the quadratic form
13004:) passing through these points. Writing the projectivization of the two quadratics as
8846:{\displaystyle \left\{{\begin{array}{l}p+u^{2}=t+v\\q=u(t-v)\\r=tv\end{array}}\right.}
497:
14182:
14096:
13967:
13926:
13841:
13832:
13703:
13514:
13449:
12862:
7914:
5707:
4893:
4080:
1786:
529:
442:
13898:
13827:
3677:{\displaystyle {\sqrt {\Delta _{1}^{2}-4\Delta _{0}^{3}}}={\sqrt {\Delta _{1}^{2}}}}
14235:
14228:
14084:
14053:
14005:
13977:
13936:
13890:
13886:
13851:
13807:
13768:
13734:
13671:
13657:
13630:
13502:
13398:
13001:
12881:
11627:
6815:
643:
480:
472:
227:
14357:
14073:
Yacoub, M.D.; Fraidenraich, G. (July 2012). "A solution to the quartic equation".
12990:
These four points are not collinear because they lie on the irreducible quadratic
14245:
14240:
14192:
14177:
14021:
13917:
13915:(1991), "The Galois theory: Equations of the second, third, and fourth degrees",
13640:
13380:
11643:
11631:
11615:
9223:
7391:
4539:
456:
case) is the highest degree such that every polynomial equation can be solved by
13830:(1954) , "Book III: On the construction of solid and supersolid problems",
14216:
14211:
13953:
13759:
Rees, E. L. (1922). "Graphical
Discussion of the Roots of a Quartic Equation".
13392:
13005:
12466:
9545:
6912:{\displaystyle \left(y^{2}+{\frac {p}{2}}\right)^{2}=-qy-r+{\frac {p^{2}}{4}}.}
4862:
2543:
are the coefficients of the second and of the first degree respectively in the
674:
476:
335:
62:
14088:
8082:
must denote the same sign, this leaves four possibilities, one for each root.
14372:
12500:
7257:{\displaystyle (-q)^{2}-4(2m)\left(m^{2}+pm+{\frac {p^{2}}{4}}-r\right)=0,\,}
517:
501:
13635:
11642:
whose roots may be variously described as a discrete
Fourier transform or a
33:"Biquadratic function" redirects here. For the use in computer science, see
14352:
7127:
5723:
For solving purposes, it is generally better to convert the quartic into a
3222:
877:
77:
were below it, or if there were no local maximum and one minimum below the
13336:. Given any two of these, their intersection has exactly the four points.
12283:{\displaystyle (s^{2}-{s_{1}}^{2})(s^{2}-{s_{2}}^{2})(s^{2}-{s_{3}}^{2}).}
11261:), it is a consequence of Vieta's formulas that their product is equal to
7485:{\displaystyle \left({\sqrt {2m}}y-{\frac {q}{2{\sqrt {2m}}}}\right)^{2}.}
7111:
which is equivalent to the original equation, whichever value is given to
1467:
is the second degree coefficient of the associated depressed quartic (see
12868:
This gives exactly the same formula for the roots as the one provided by
10599:
10107:
9373:
8761:
8581:
651:
51:
14110:
13956:(1984) , "Of a new method of resolving equations of the fourth degree",
6814:
This depressed quartic can be solved by means of a method discovered by
5507:{\displaystyle P(x)=a_{0}x^{4}+a_{1}x^{3}+a_{2}x^{2}+a_{1}mx+a_{0}m^{2}}
4836:{\displaystyle Q(x)=(c_{2}x^{2}+c_{1}x+c_{0})(d_{2}x^{2}+d_{1}x+d_{0}).}
1390:
This may be refined by considering the signs of four other polynomials:
14156:
14114:
14065:
14017:
13780:
13746:
13683:
590:
513:
331:
211:
26:
This article is about the univariate case. For the bivariate case, see
1570:
is the first degree coefficient of the associated depressed quartic;
14286:
13796:"Quantifier elimination: Optimal solution for two classical examples"
13547:"DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces, p. 36"
3947:
14057:
14009:
13833:
The
Geometry of Rene Descartes with a facsimile of the first edition
13772:
13738:
13675:
1869:≠ 0)), there are a real double root and two complex conjugate roots.
1818:> 0 then there are two pairs of non-real complex conjugate roots.
14276:
4475:{\displaystyle Q(x)=a_{4}x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}.}
3886:
This is always possible except if the quartic may be factored into
504:
of the roots of polynomials, of which this theorem was one result.
457:
434:
4685:{\displaystyle Q(x)=(x-x_{1})(b_{3}x^{3}+b_{2}x^{2}+b_{1}x+b_{0})}
2056:
1764:{\displaystyle D=64a^{3}e-16a^{2}c^{2}+16ab^{2}c-16a^{2}bd-3b^{4}}
85:
axis, or if there were no local maximum and one minimum above the
11458:
This argument suggests another way of choosing the square roots:
6944:
to both sides. After regrouping the coefficients of the power of
623:
552:
94:
74:
66:
42:
12910:
coordinates of the intersections of the two quadratic equations
4079:
at least three roots are equal to each other, and the roots are
876:
the nature of its roots is mainly determined by the sign of its
12950:
that two quadratics intersect in four points is an instance of
12499:
These expressions are unnecessarily complicated, involving the
11556:{\displaystyle -{\frac {q}{{\sqrt {\alpha }}{\sqrt {\beta }}}}}
5845:{\displaystyle a_{4}x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}=0}
1796:
then either the equation's four roots are all real or none is.
1777:
The possible cases for the nature of the roots are as follows:
541:
13207:
for any point in the projective line — in other words, where
12459:
This polynomial is of degree six, but only of degree three in
9869:) and this suggests that the roots of that cubic are equal to
9222:, then solving this equation becomes finding the roots of the
8085:
Therefore, the solutions of the original quartic equation are
7913:
This equation is easily solved by applying to each factor the
4542:
coefficients (or more generally with coefficients in the same
3974:
is biquadratic; it may thus be solved by the method described
3194:{\displaystyle \Delta _{1}^{2}-4\Delta _{0}^{3}=-27\Delta \ ,}
14042:
Carpenter, W. (1966). "On the solution of the real quartic".
7406:
is a root of this equation, the right-hand side of equation (
6741:
As explained in the preceding section, we may start with the
564:
A quartic equation arises also in the process of solving the
521:
14072:
12861:
and then solving for the roots of the two factors using the
11208:
from the previous equalities. Then, one computes the number
9515:) has a non-zero root which is the square of a rational, or
7516:. This is always possible except for the depressed equation
551:, the torus is a shape that is commonly associated with the
13660:(1998), "Reflections on Reflection in a Spherical Mirror",
11034:
10777:
10387:
9478:
8840:
8672:
4868:
Detecting the existence of such factorizations can be done
12845:{\displaystyle F_{1}(x)\times F_{2}(x)=x^{4}+cx^{2}+dx+e.}
4110:
is a common root of the quartic and its second derivative
7122:
may be arbitrarily chosen, we will choose it in order to
1884:≠ 0, there are a triple root and a simple root, all real.
1850:, there are a real double root and two real simple roots.
500:
prior to dying in a duel in 1832 later led to an elegant
14155:
13000:
and thus there is a 1-parameter family of quadratics (a
4173:
it is thus also the unique root of the remainder of the
13383: – Linear map or polynomial function of degree one
10056:
are the roots of the resolvent cubic, then the numbers
4920:
In fact, several methods of solving quartic equations (
13229:
12000:
11909:
11818:
11727:
9503:
merely exchanges the two quadratics with one another.
8689:
3971:
3906:
2544:
2279:
1785:
then the equation has two distinct real roots and two
1468:
13228:
13017:
12753:
12526:
12469:. By substituting the roots in the expression of the
12352:
12311:, this polynomial may be expanded in a polynomial in
12162:
11704:
11526:
10802:
10593:
10101:
9367:
9236:
8873:
8755:
8575:
8566:, this results in the following system of equations:
8293:
8094:
7926:
7733:
7554:
7423:
7278:
7151:
6959:
6827:
6753:
6074:
5740:
5380:
5184:
4966:
4704:
4566:
4358:
4301:
4275:
4210:
4183:
4116:
4089:
4049:
4013:
3987:
3956:
3892:
3866:
3846:
3817:
3785:
3755:
3719:
3690:
3611:
3575:
3549:
3465:
3355:
3318:
3298:
3278:
3249:
3207:
3136:
2952:
2723:
2555:
2299:
2189:
2066:
1989:
1652:
1579:
1480:
1399:
888:
789:
686:
629:
604:
The characteristic equation of a fourth-order linear
366:
240:
114:
10403:
It is a consequence of the first two equations that
7367:{\displaystyle 8m^{3}+8pm^{2}+(2p^{2}-8r)m-q^{2}=0.}
5718:
4557:). Such a factorization will take one of two forms:
3939:{\displaystyle \left(x+{\tfrac {b}{4a}}\right)^{4}.}
3840:
then one must change the choice of the cube root in
12436:{\displaystyle s^{6}+2cs^{4}+(c^{2}-4e)s^{2}-d^{2}}
10000:. (Of course, this also follows from the fact that
1831:root. Here are the different cases that can occur:
597:are the roots of a quartic polynomial which is the
524:. It follows that quartic equations often arise in
13723:Aude, H. T. R. (1949), "Notes on Quartic Curves",
13506:
13255:
13180:
12875:
12855:We therefore can solve the quartic by solving for
12844:
12733:
12435:
12282:
12075:
11606:, in which case there is a much simpler approach.
11555:
11039:
10782:
10392:
9483:
9315:
9201:
8845:
8688:This can be simplified by starting again with the
8677:
8551:
8245:
8042:
7902:
7664:
7484:
7366:
7256:
7087:
6911:
6803:
6649:
5855:be the general quartic equation we want to solve.
5844:
5506:
5355:
5040:
4835:
4684:
4474:
4335:and no cube root is needed to represent the roots.
4323:
4287:
4259:
4196:
4165:
4102:
4071:
4035:
3999:
3962:
3938:
3878:
3852:
3832:
3801:
3771:
3741:
3705:
3676:
3597:
3561:
3529:
3445:
3327:
3312:is also real, despite being expressed in terms of
3304:
3284:
3264:
3213:
3193:
3116:
2911:
2701:
2524:
2259:
2132:
2040:
1909:= 0, there are two complex conjugate double roots.
1763:
1632:
1536:
1433:
1379:
858:
755:
422:
312:
195:
13592:
13473:
12099:in exactly the same way. Since we know the value
9548:. Unlike the previous methods, both of which use
6948:on the right-hand side, this gives the equation
2937:
1807:< 0 then all four roots are real and distinct.
1643:which is 0 if the quartic has a triple root; and
14370:
11609:
9316:{\displaystyle U^{3}+2pU^{2}+(p^{2}-4r)U-q^{2},}
6926:into the factor on the left-hand side by adding
6046:gives, after regrouping the terms, the equation
5041:{\displaystyle Q(x)=a_{4}x^{4}+a_{2}x^{2}+a_{0}}
1774:which is 0 if the quartic has two double roots.
646:of the graph of a quartic function, and letting
612:is a quartic equation. An example arises in the
13911:
11298:. But a straightforward computation shows that
3237:
196:{\displaystyle f(x)=ax^{4}+bx^{3}+cx^{2}+dx+e,}
14178:Zero polynomial (degree undefined or −1 or −∞)
13867:
12090:we may express the roots in terms of the four
8257:
7917:. Solving them we may write the four roots as
7126:on the right-hand side. This implies that the
4932:) are based upon finding such factorizations.
14141:
14041:
13620:
13246:
13233:
5366:
2260:{\displaystyle ax^{4}+bx^{3}+cx^{2}+dx+e=0\,}
12869:
12332:fundamental theorem of symmetric polynomials
9861:is one of the roots of the resolvent cubic (
7495:However, this induces a division by zero if
4925:
4896:) then there is always such a factorization;
2051:
2041:{\displaystyle 16a^{2}\Delta _{0}=3D+P^{2};}
584:distance of closest approach of two ellipses
313:{\displaystyle ax^{4}+bx^{3}+cx^{2}+dx+e=0,}
13870:"Factoring quartic polynomials: A lost art"
11602:, that is, only when we are dealing with a
9544:A variant of the previous method is due to
4921:
2275:
1633:{\displaystyle \Delta _{0}=c^{2}-3bd+12ae,}
859:{\displaystyle ax^{4}+bx^{3}+cx^{2}+dx+e=0}
14148:
14134:
13861:
13536:(Chapman & Hall/CRC Mathematics, 2004)
13256:{\displaystyle \textstyle {\binom {4}{2}}}
11085:}, but, in fact, it provides no more than
9965:. This is indeed true and it follows from
7530:is a root of the cubic equation such that
6669:is a root of this depressed quartic, then
2133:{\displaystyle x^{4}+ax^{3}+bx^{2}+cx+d=0}
1967:≤ 0 is not one of the cases. In fact, if
1827:then (and only then) the polynomial has a
738:
46:Graph of a polynomial of degree 4, with 3
13826:
13811:
13702:. Springer Science & Business Media.
13634:
13389: – Polynomial function of degree two
11603:
7253:
4929:
3975:
3742:{\displaystyle {\sqrt {\Delta _{1}^{2}}}}
2256:
65:roots). If one or the other of the local
14111:Quartic formula as four single equations
13993:
13395: – Polynomial function of degree 3
13371:and corresponds to the resolvent cubic.
12887:The four roots of the depressed quartic
12153:. These are the roots of the polynomial
9532:; this can readily be checked using the
8709:, which can be obtained by substituting
2280:§ Converting to a depressed quartic
2055:
1898:< 0, there are two real double roots.
41:
13695:
13656:
13604:MacTutor History of Mathematics Archive
13586:
13501:
13485:MacTutor History of Mathematics Archive
13401: – Polynomial function of degree 5
4935:
14371:
13793:
13426:as opposed to its conventional use as
12503:, which can be avoided as follows. If
11567:Of course, this will make no sense if
8279:
7394:of the quartic equation. The value of
3970:is zero, in which case the associated
775:
650:be the intersection of the inflection
626:can be found using quartic equations.
14129:
13952:
13566:
13447:
11172:and uses them to compute the numbers
6736:
4344:
4083:of the coefficients. The triple root
2048:so this combination is not possible.
1537:{\displaystyle R=b^{3}+8da^{2}-4abc,}
622:between spheres, cylinders, or other
586:involves solving a quartic equation.
423:{\displaystyle f(x)=ax^{4}+cx^{2}+e.}
13758:
13722:
13699:Theory of Vibration: An Introduction
13650:
12343:
12086:then since the transformation is an
9227:
7269:
6950:
6804:{\displaystyle y^{4}+py^{2}+qy+r=0.}
5059:, which is easy to solve as follows
2278:by back changing the variables (see
579:" This leads to a quartic equation.
9539:
5867:, provides the equivalent equation
5713:
3225:. For the cube root expression for
2938:§ Special cases of the formula
780:Given the general quartic equation
13:
14035:
13696:Shabana, A. A. (8 December 1995).
13430:(except when otherwise specified).
13418:For the purposes of this article,
13411:
13237:
12954:. Explicitly, the four points are
11650:for the general method. Denote by
4538:are non-constant polynomials with
4324:{\displaystyle \Delta _{0}\neq 0,}
4303:
4276:
4260:{\displaystyle x_{1}+3x_{0}=-b/a.}
4051:
4015:
3988:
3787:
3757:
3723:
3658:
3636:
3615:
3577:
3550:
3501:
3485:
3414:
3250:
3208:
3182:
3159:
3138:
3013:
2958:
2877:
2856:
2841:
2801:
2004:
1581:
893:
630:Inflection points and golden ratio
516:of the intersection points of two
210:is nonzero, which is defined by a
16:Polynomial function of degree four
14:
14400:
14104:
13761:The American Mathematical Monthly
13599:"Abu Ali al-Hasan ibn al-Haytham"
13509:Ars magna or The Rules of Algebra
13191:the pencil is given by the forms
12341:, this results in the polynomial
5719:Converting to a depressed quartic
4166:{\displaystyle 2(6ax^{2}+3bx+c);}
2282:) and using the formulas for the
2180:for the general quartic equation
13913:van der Waerden, Bartel Leendert
9358:, which is trivially factored),
5164:. Then the roots of the quartic
5055:; equating it to zero defines a
4339:
13987:
13946:
13905:
13820:
13800:Journal of Symbolic Computation
13787:
13752:
13716:
13689:
12876:Solving with algebraic geometry
12302:by their values in term of the
528:and all related fields such as
507:
13921:, vol. 1 (7th ed.),
13891:10.1080/0025570X.2007.11953453
13614:
13560:
13539:
13526:
13495:
13467:
13441:
13413:
13145:
13127:
13050:
13032:
12792:
12786:
12770:
12764:
12647:
12641:
12547:
12541:
12407:
12385:
12274:
12239:
12236:
12201:
12198:
12163:
12126:, we only need the values for
12063:
12011:
11972:
11920:
11881:
11829:
11790:
11738:
10366:
10340:
10337:
10311:
10295:
10269:
10266:
10240:
10224:
10198:
10195:
10169:
9525:is the square of rational and
9291:
9269:
9137:
9128:
9125:
9116:
9093:
9090:
9087:
9075:
9060:
9057:
9054:
9042:
9027:
9024:
8995:
8982:
8960:
8947:
8908:
8888:
8817:
8805:
8530:
8512:
8496:
8475:
8459:
8447:
8421:
8393:
8390:
8362:
7339:
7314:
7186:
7177:
7162:
7152:
6922:Then, we introduce a variable
5706:may be solved by applying the
5390:
5384:
4976:
4970:
4827:
4775:
4772:
4720:
4714:
4708:
4679:
4604:
4601:
4582:
4576:
4570:
4368:
4362:
4157:
4120:
4072:{\displaystyle \Delta _{1}=0,}
4036:{\displaystyle \Delta _{0}=0,}
3598:{\displaystyle \Delta _{0}=0,}
1920:, all four roots are equal to
747:
739:
376:
370:
124:
118:
1:
14338:Horner's method of evaluation
13997:American Mathematical Monthly
13813:10.1016/S0747-7171(88)80015-4
13726:American Mathematical Monthly
13663:American Mathematical Monthly
13434:
13267:different ways. Denote these
12940:i.e., using the substitution
12906:may also be expressed as the
11610:Solving by Lagrange resolvent
7685:, which can be rearranged as
7675:This equation is of the form
4870:using the resolvent cubic of
4846:In either case, the roots of
4349:Consider the general quartic
3562:{\displaystyle \Delta \neq 0}
3265:{\displaystyle \Delta >0,}
35:Biquadratic rational function
13534:Galois Theory, Third Edition
11646:transform of the roots; see
10538:is the other square root of
10490:is the other square root of
10441:is the other square root of
8075:. As the two occurrences of
7380:
3802:{\displaystyle \Delta _{1}.}
3772:{\displaystyle \Delta _{1},}
3238:Special cases of the formula
2545:associated depressed quartic
1434:{\displaystyle P=8ac-3b^{2}}
549:computer-aided manufacturing
538:computer-aided manufacturing
7:
14343:Polynomial identity testing
13374:
12512:
12449:
11590:
11256:
9864:
9510:
9329:
8856:One can now eliminate both
8726:. Since the coefficient of
7719:
7540:
7409:
7101:
5582:). The change of variables
5062:Let the auxiliary variable
4928:, and, to a lesser extent,
869:with real coefficients and
770:
657:and the quartic, nearer to
334:of a quartic function is a
10:
14405:
12882:reducible quadratic curves
7398:may thus be obtained from
7267:which may be rewritten as
7142:is a root of the equation
6743:depressed quartic equation
5367:Quasi-palindromic equation
3713:that is one should define
614:Timoshenko-Rayleigh theory
467:
32:
25:
18:
14315:
14254:
14167:
14089:10.1017/s002555720000454x
13569:"Crossed Ladders Problem"
12509:is any non-zero root of (
11626:on four elements has the
4288:{\displaystyle \Delta =0}
4000:{\displaystyle \Delta =0}
3684:has to be chosen to have
2052:General formula for roots
599:characteristic polynomial
13609:University of St Andrews
13490:University of St Andrews
13405:
11634:. This suggests using a
8864:by doing the following:
3879:{\displaystyle S\neq 0.}
3779:maintaining the sign of
3706:{\displaystyle Q\neq 0,}
19:Not to be confused with
14328:Greatest common divisor
13868:Brookfield, G. (2007).
12317:whose coefficients are
10548:Therefore, the numbers
10447:. For the same reason,
9499:for the square root of
7724:) may be rewritten as
7716:. Therefore, equation (
5695:, the quartic equation
4899:if we are working over
4886:if we are working over
4546:as the coefficients of
3341:trigonometric functions
3214:{\displaystyle \Delta }
566:crossed ladders problem
14200:Quadratic function (2)
13257:
13182:
12846:
12735:
12437:
12284:
12077:
11557:
11041:
10784:
10394:
9485:
9317:
9203:
8847:
8679:
8553:
8256:A comparison with the
8247:
8044:
7904:
7666:
7486:
7368:
7258:
7089:
6913:
6805:
6651:
5846:
5575:(it is palindromic if
5508:
5357:
5042:
4837:
4686:
4476:
4325:
4289:
4261:
4198:
4167:
4104:
4073:
4037:
4001:
3964:
3940:
3880:
3854:
3834:
3803:
3773:
3743:
3707:
3678:
3599:
3563:
3531:
3447:
3329:
3306:
3286:
3266:
3221:is the aforementioned
3215:
3195:
3118:
2913:
2703:
2526:
2261:
2141:
2134:
2042:
1765:
1634:
1538:
1435:
1381:
860:
757:
526:computational geometry
424:
314:
197:
90:
73:axis, or if the local
14183:Constant function (0)
14121:Ferrari's achievement
13636:10.1214/ss/1009212817
13573:mathworld.wolfram.com
13454:mathworld.wolfram.com
13258:
13183:
12847:
12736:
12438:
12319:symmetric polynomials
12285:
12078:
11558:
11042:
10785:
10395:
9486:
9318:
9204:
8848:
8680:
8564:equating coefficients
8554:
8248:
8045:
7905:
7667:
7487:
7369:
7259:
7090:
6914:
6806:
6652:
5847:
5509:
5358:
5043:
4882:. It turns out that:
4838:
4687:
4477:
4326:
4290:
4262:
4199:
4197:{\displaystyle x_{1}}
4168:
4105:
4103:{\displaystyle x_{0}}
4074:
4038:
4002:
3965:
3941:
3881:
3855:
3835:
3804:
3774:
3744:
3708:
3679:
3600:
3564:
3532:
3448:
3330:
3307:
3287:
3267:
3216:
3196:
3119:
2914:
2704:
2527:
2262:
2135:
2059:
2043:
1766:
1635:
1539:
1436:
1382:
861:
758:
610:differential equation
534:computer-aided design
425:
315:
198:
45:
14389:Polynomial functions
14316:Tools and algorithms
14236:Quintic function (5)
14224:Quartic function (4)
14161:polynomial functions
14076:Mathematical Gazette
14045:Mathematics Magazine
13878:Mathematics Magazine
13595:Robertson, Edmund F.
13476:Robertson, Edmund F.
13226:
13015:
13008:in three variables:
12751:
12524:
12501:cubic roots of unity
12350:
12160:
11702:
11673:, the four roots of
11604:biquadratic equation
11524:
11054:choices for the set
10800:
10591:
10514:is a square root of
10466:is a square root of
10419:is a square root of
10099:
9365:
9234:
8871:
8753:
8573:
8291:
8092:
7924:
7731:
7552:
7421:
7276:
7149:
6957:
6825:
6751:
6072:
5738:
5378:
5182:
5057:biquadratic equation
5053:biquadratic function
4964:
4936:Biquadratic equation
4702:
4564:
4356:
4299:
4273:
4208:
4204:can be deduced from
4181:
4114:
4087:
4047:
4011:
3985:
3954:
3890:
3864:
3844:
3833:{\displaystyle S=0,}
3815:
3783:
3753:
3717:
3688:
3609:
3573:
3547:
3463:
3353:
3316:
3296:
3276:
3247:
3205:
3134:
2950:
2721:
2553:
2297:
2187:
2064:
1987:
1650:
1577:
1478:
1397:
886:
787:
684:
494:Abel–Ruffini theorem
462:Abel–Ruffini theorem
364:
351:biquadratic function
238:
112:
14246:Septic equation (7)
14241:Sextic equation (6)
14188:Linear function (1)
13959:Elements of Algebra
13794:Lazard, D. (1988).
13623:Statistical Science
13593:O'Connor, John J.;
13567:Weisstein, Eric W.
13474:O'Connor, John J.;
13448:Weisstein, Eric W.
12978:for the four roots
11648:Lagrange resolvents
11267:and therefore that
9570:. Observe that, if
8280:Descartes' solution
7124:complete the square
4957:then the function
3736:
3671:
3649:
3628:
3514:
3337:casus irreducibilis
3172:
3151:
2890:
2869:
776:Nature of the roots
606:difference equation
460:, according to the
345:is used instead of
341:Sometimes the term
61:axis) (and thus no
28:Quartic plane curve
14379:Elementary algebra
14212:Cubic function (3)
14205:Quadratic equation
13480:"Lodovico Ferrari"
13450:"Quartic Equation"
13387:Quadratic function
13253:
13252:
13178:
13176:
12842:
12731:
12729:
12433:
12280:
12073:
12071:
12009:
11918:
11827:
11736:
11553:
11408:If this number is
11253:are the roots of (
11120:} becomes the set
11037:
11032:
10780:
10775:
10390:
10385:
9829:. In other words,
9534:rational root test
9481:
9476:
9313:
9199:
9197:
8843:
8838:
8675:
8670:
8549:
8547:
8243:
8040:
7900:
7662:
7482:
7364:
7254:
7136:quadratic equation
7085:
6909:
6801:
6737:Ferrari's solution
6647:
6645:
5842:
5642:quadratic equation
5504:
5353:
5351:
5038:
4859:quadratic function
4833:
4682:
4472:
4345:Reducible quartics
4321:
4285:
4257:
4194:
4175:Euclidean division
4163:
4100:
4081:rational functions
4069:
4033:
3997:
3960:
3936:
3920:
3876:
3850:
3830:
3799:
3769:
3739:
3722:
3703:
3674:
3657:
3635:
3614:
3595:
3559:
3527:
3500:
3443:
3328:{\displaystyle Q;}
3325:
3302:
3282:
3262:
3231:cubic counterparts
3211:
3191:
3158:
3137:
3114:
3112:
2909:
2907:
2876:
2855:
2699:
2697:
2522:
2520:
2257:
2142:
2130:
2038:
1761:
1630:
1534:
1431:
1377:
1375:
856:
753:
420:
355:quadratic function
310:
220:quartic polynomial
193:
91:
57:(crossings of the
14366:
14365:
14307:Quasi-homogeneous
13973:978-1-4613-8511-0
13709:978-0-387-94524-8
13658:Neumann, Peter M.
13503:Cardano, Gerolamo
13244:
12870:Descartes' method
12863:quadratic formula
12725:
12707:
12687:
12625:
12607:
12587:
12517:), and if we set
12457:
12456:
12293:Substituting the
12008:
11917:
11826:
11735:
11551:
11548:
11541:
11028:
11023:
11017:
11007:
10997:
10967:
10961:
10951:
10941:
10911:
10905:
10895:
10885:
10858:
10852:
10842:
10832:
10771:
10766:
10729:
10692:
10381:
10038:.) Therefore, if
9670:are the roots of
9636:are the roots of
9337:
9336:
8690:depressed quartic
8260:above shows that
8238:
8232:
8225:
8223:
8212:
8157:
8131:
8035:
8029:
8022:
8020:
8009:
7954:
7915:quadratic formula
7887:
7884:
7859:
7840:
7807:
7804:
7779:
7760:
7646:
7643:
7621:
7582:
7466:
7463:
7438:
7400:Cardano's formula
7388:
7387:
7231:
7138:is zero, that is
7109:
7108:
7074:
6987:
6904:
6855:
6638:
6455:
6376:
6253:
6199:
6118:
5725:depressed quartic
5708:quadratic formula
5344:
5303:
5262:
5221:
4926:Descartes' method
4894:real closed field
3972:depressed quartic
3963:{\displaystyle q}
3919:
3860:in order to have
3853:{\displaystyle Q}
3737:
3672:
3650:
3518:
3515:
3441:
3439:
3423:
3409:
3389:
3385:
3370:
3305:{\displaystyle S}
3285:{\displaystyle Q}
3187:
2903:
2897:
2891:
2820:
2813:
2785:
2765:
2761:
2746:
2693:
2614:
2516:
2514:
2474:
2455:
2431:
2407:
2405:
2365:
2346:
2322:
1787:complex conjugate
745:
730:
724:
705:
644:inflection points
616:of beam bending.
573:Alhazen's problem
530:computer graphics
452:The degree four (
14396:
14229:Quartic equation
14150:
14143:
14136:
14127:
14126:
14100:
14069:
14029:
14028:
13991:
13985:
13984:
13950:
13944:
13943:
13909:
13903:
13902:
13874:
13865:
13859:
13858:
13824:
13818:
13817:
13815:
13806:(1–2): 261–266.
13791:
13785:
13784:
13756:
13750:
13749:
13720:
13714:
13713:
13693:
13687:
13686:
13654:
13648:
13647:
13638:
13618:
13612:
13611:
13590:
13584:
13583:
13581:
13579:
13564:
13558:
13557:
13551:
13543:
13537:
13530:
13524:
13523:
13512:
13499:
13493:
13492:
13471:
13465:
13464:
13462:
13460:
13445:
13417:
13399:Quintic function
13370:
13364:
13358:
13354:
13335:
13312:
13289:
13266:
13262:
13260:
13259:
13254:
13251:
13250:
13249:
13236:
13218:
13212:
13206:
13187:
13185:
13184:
13179:
13177:
13173:
13172:
13126:
13125:
13109:
13108:
13069:
13068:
13031:
13030:
13002:pencil of curves
12999:
12987:of the quartic.
12986:
12977:
12952:Bézout's theorem
12949:
12939:
12928:
12909:
12905:
12860:
12851:
12849:
12848:
12843:
12823:
12822:
12807:
12806:
12785:
12784:
12763:
12762:
12740:
12738:
12737:
12732:
12730:
12726:
12724:
12713:
12708:
12703:
12702:
12693:
12688:
12680:
12666:
12665:
12640:
12639:
12626:
12624:
12613:
12608:
12603:
12602:
12593:
12588:
12580:
12566:
12565:
12540:
12539:
12508:
12495:
12486:
12478:in terms of the
12477:
12464:
12451:
12442:
12440:
12439:
12434:
12432:
12431:
12419:
12418:
12397:
12396:
12381:
12380:
12362:
12361:
12344:
12340:
12329:
12316:
12310:
12301:
12289:
12287:
12286:
12281:
12273:
12272:
12267:
12266:
12265:
12251:
12250:
12235:
12234:
12229:
12228:
12227:
12213:
12212:
12197:
12196:
12191:
12190:
12189:
12175:
12174:
12152:
12143:
12134:
12125:
12124:
12122:
12121:
12118:
12115:
12098:
12082:
12080:
12079:
12074:
12072:
12062:
12061:
12049:
12048:
12036:
12035:
12023:
12022:
12010:
12001:
11991:
11990:
11971:
11970:
11958:
11957:
11945:
11944:
11932:
11931:
11919:
11910:
11900:
11899:
11880:
11879:
11867:
11866:
11854:
11853:
11841:
11840:
11828:
11819:
11809:
11808:
11789:
11788:
11776:
11775:
11763:
11762:
11750:
11749:
11737:
11728:
11718:
11717:
11694:
11672:
11668:
11664:
11658:
11640:
11639:
11628:Klein four-group
11625:
11601:
11586:
11582:
11578:
11572:
11562:
11560:
11559:
11554:
11552:
11550:
11549:
11544:
11542:
11537:
11531:
11519:
11518:
11517:
11503:
11497:
11496:
11495:
11482:
11476:
11475:
11474:
11454:
11444:
11434:
11424:
11414:
11404:
11326:
11325:
11318:
11317:
11310:
11309:
11297:
11292:
11291:
11284:
11283:
11276:
11275:
11266:
11252:
11246:
11240:
11234:
11233:
11232:
11225:
11224:
11217:
11216:
11207:
11198:
11189:
11180:
11171:
11165:
11159:
11150:
11119:
11088:
11084:
11053:
11046:
11044:
11043:
11038:
11036:
11033:
11029:
11026:
11024:
11019:
11018:
11013:
11008:
11003:
10998:
10993:
10987:
10982:
10981:
10968:
10963:
10962:
10957:
10952:
10947:
10942:
10937:
10931:
10926:
10925:
10912:
10907:
10906:
10901:
10896:
10891:
10886:
10881:
10878:
10873:
10872:
10859:
10854:
10853:
10848:
10843:
10838:
10833:
10828:
10825:
10820:
10819:
10789:
10787:
10786:
10781:
10779:
10776:
10772:
10769:
10767:
10762:
10757:
10756:
10744:
10743:
10730:
10725:
10720:
10719:
10707:
10706:
10693:
10688:
10683:
10682:
10670:
10669:
10650:
10649:
10637:
10636:
10624:
10623:
10611:
10610:
10583:
10574:
10565:
10556:
10543:
10537:
10519:
10513:
10495:
10489:
10471:
10465:
10446:
10440:
10424:
10418:
10399:
10397:
10396:
10391:
10389:
10386:
10382:
10379:
10365:
10364:
10352:
10351:
10336:
10335:
10323:
10322:
10294:
10293:
10281:
10280:
10265:
10264:
10252:
10251:
10223:
10222:
10210:
10209:
10194:
10193:
10181:
10180:
10158:
10157:
10145:
10144:
10132:
10131:
10119:
10118:
10091:
10082:
10073:
10064:
10055:
10049:
10043:
10037:
9999:
9967:Vieta's formulas
9964:
9932:
9900:
9860:
9828:
9789:
9767:
9745:
9736:
9727:
9718:
9709:
9683:
9669:
9660:
9649:
9635:
9626:
9615:
9569:
9540:Euler's solution
9531:
9524:
9502:
9498:
9490:
9488:
9487:
9482:
9480:
9477:
9470:
9459:
9458:
9427:
9416:
9415:
9357:
9351:
9331:
9322:
9320:
9319:
9314:
9309:
9308:
9281:
9280:
9265:
9264:
9246:
9245:
9228:
9221:
9208:
9206:
9205:
9200:
9198:
9191:
9190:
9172:
9162:
9161:
9143:
9115:
9114:
9099:
9023:
9022:
9007:
9003:
9002:
8981:
8980:
8968:
8967:
8946:
8945:
8929:
8928:
8916:
8915:
8906:
8905:
8887:
8886:
8863:
8859:
8852:
8850:
8849:
8844:
8842:
8839:
8779:
8778:
8745:
8735:
8731:
8725:
8719:
8708:
8684:
8682:
8681:
8676:
8674:
8671:
8558:
8556:
8555:
8550:
8548:
8508:
8507:
8471:
8470:
8443:
8442:
8427:
8405:
8404:
8374:
8373:
8339:
8338:
8323:
8322:
8307:
8306:
8275:
8270:
8269:
8252:
8250:
8249:
8244:
8239:
8234:
8233:
8231:
8227:
8226:
8224:
8219:
8217:
8213:
8208:
8205:
8203:
8202:
8170:
8168:
8167:
8158:
8150:
8148:
8147:
8137:
8132:
8130:
8129:
8128:
8115:
8114:
8105:
8081:
8074:
8070:
8066:
8059:
8049:
8047:
8046:
8041:
8036:
8031:
8030:
8028:
8024:
8023:
8021:
8016:
8014:
8010:
8005:
8002:
8000:
7999:
7967:
7965:
7964:
7955:
7947:
7945:
7944:
7934:
7909:
7907:
7906:
7901:
7893:
7889:
7888:
7886:
7885:
7877:
7868:
7860:
7852:
7841:
7833:
7828:
7827:
7813:
7809:
7808:
7806:
7805:
7797:
7788:
7780:
7772:
7761:
7753:
7748:
7747:
7715:
7695:
7684:
7671:
7669:
7668:
7663:
7658:
7657:
7652:
7648:
7647:
7645:
7644:
7636:
7627:
7622:
7614:
7600:
7599:
7594:
7590:
7583:
7575:
7570:
7569:
7536:
7529:
7522:
7515:
7508:
7501:
7491:
7489:
7488:
7483:
7478:
7477:
7472:
7468:
7467:
7465:
7464:
7456:
7447:
7439:
7431:
7414:) is the square
7405:
7397:
7382:
7373:
7371:
7370:
7365:
7357:
7356:
7329:
7328:
7310:
7309:
7291:
7290:
7270:
7263:
7261:
7260:
7255:
7243:
7239:
7232:
7227:
7226:
7217:
7203:
7202:
7170:
7169:
7141:
7133:
7121:
7118:As the value of
7114:
7103:
7094:
7092:
7091:
7086:
7075:
7070:
7069:
7060:
7046:
7045:
7024:
7023:
7005:
7004:
6999:
6995:
6988:
6980:
6975:
6974:
6951:
6947:
6943:
6925:
6918:
6916:
6915:
6910:
6905:
6900:
6899:
6890:
6867:
6866:
6861:
6857:
6856:
6848:
6843:
6842:
6816:Lodovico Ferrari
6810:
6808:
6807:
6802:
6779:
6778:
6763:
6762:
6732:
6730:
6728:
6727:
6718:
6715:
6695:
6694:
6692:
6691:
6688:
6685:
6668:
6656:
6654:
6653:
6648:
6646:
6639:
6637:
6636:
6635:
6630:
6629:
6628:
6613:
6612:
6611:
6602:
6601:
6596:
6595:
6594:
6583:
6582:
6567:
6566:
6561:
6560:
6559:
6548:
6547:
6538:
6537:
6522:
6521:
6516:
6515:
6514:
6503:
6502:
6487:
6486:
6481:
6480:
6479:
6461:
6456:
6451:
6447:
6446:
6410:
6409:
6393:
6377:
6375:
6374:
6373:
6368:
6367:
6366:
6351:
6350:
6349:
6344:
6343:
6342:
6331:
6330:
6315:
6314:
6305:
6304:
6295:
6294:
6279:
6278:
6273:
6272:
6271:
6259:
6254:
6249:
6227:
6226:
6216:
6200:
6198:
6197:
6196:
6191:
6190:
6189:
6174:
6173:
6172:
6167:
6166:
6165:
6148:
6147:
6138:
6137:
6124:
6119:
6114:
6113:
6112:
6090:
6064:
6045:
6041:
6040:
6038:
6037:
6034:
6031:
6017:
6016:
6014:
6013:
6005:
6002:
5985:
5984:
5982:
5981:
5973:
5970:
5953:
5952:
5950:
5949:
5941:
5938:
5921:
5920:
5918:
5917:
5909:
5906:
5889:
5866:
5851:
5849:
5848:
5843:
5835:
5834:
5819:
5818:
5806:
5805:
5796:
5795:
5783:
5782:
5773:
5772:
5760:
5759:
5750:
5749:
5714:Solution methods
5705:
5694:
5679:
5639:
5637:
5635:
5634:
5629:
5626:
5611:
5610:
5608:
5607:
5602:
5599:
5581:
5574:
5572:
5570:
5569:
5564:
5561:
5549:
5547:
5546:
5541:
5538:
5513:
5511:
5510:
5505:
5503:
5502:
5493:
5492:
5474:
5473:
5461:
5460:
5451:
5450:
5438:
5437:
5428:
5427:
5415:
5414:
5405:
5404:
5362:
5360:
5359:
5354:
5352:
5345:
5343:
5342:
5333:
5321:
5320:
5304:
5302:
5301:
5292:
5280:
5279:
5263:
5261:
5260:
5251:
5239:
5238:
5222:
5220:
5219:
5210:
5198:
5197:
5174:
5163:
5153:be the roots of
5152:
5143:
5134:
5097:
5091:
5082:
5071:
5047:
5045:
5044:
5039:
5037:
5036:
5024:
5023:
5014:
5013:
5001:
5000:
4991:
4990:
4956:
4922:Ferrari's method
4915:
4904:
4891:
4880:
4856:
4842:
4840:
4839:
4834:
4826:
4825:
4810:
4809:
4797:
4796:
4787:
4786:
4771:
4770:
4755:
4754:
4742:
4741:
4732:
4731:
4691:
4689:
4688:
4683:
4678:
4677:
4662:
4661:
4649:
4648:
4639:
4638:
4626:
4625:
4616:
4615:
4600:
4599:
4556:
4537:
4526:
4515:
4481:
4479:
4478:
4473:
4468:
4467:
4452:
4451:
4439:
4438:
4429:
4428:
4416:
4415:
4406:
4405:
4393:
4392:
4383:
4382:
4330:
4328:
4327:
4322:
4311:
4310:
4294:
4292:
4291:
4286:
4266:
4264:
4263:
4258:
4250:
4236:
4235:
4220:
4219:
4203:
4201:
4200:
4195:
4193:
4192:
4172:
4170:
4169:
4164:
4138:
4137:
4109:
4107:
4106:
4101:
4099:
4098:
4078:
4076:
4075:
4070:
4059:
4058:
4042:
4040:
4039:
4034:
4023:
4022:
4006:
4004:
4003:
3998:
3969:
3967:
3966:
3961:
3945:
3943:
3942:
3937:
3932:
3931:
3926:
3922:
3921:
3918:
3907:
3885:
3883:
3882:
3877:
3859:
3857:
3856:
3851:
3839:
3837:
3836:
3831:
3808:
3806:
3805:
3800:
3795:
3794:
3778:
3776:
3775:
3770:
3765:
3764:
3748:
3746:
3745:
3740:
3738:
3735:
3730:
3721:
3712:
3710:
3709:
3704:
3683:
3681:
3680:
3675:
3673:
3670:
3665:
3656:
3651:
3648:
3643:
3627:
3622:
3613:
3604:
3602:
3601:
3596:
3585:
3584:
3568:
3566:
3565:
3560:
3536:
3534:
3533:
3528:
3523:
3519:
3517:
3516:
3513:
3508:
3499:
3493:
3492:
3483:
3452:
3450:
3449:
3444:
3442:
3440:
3432:
3424:
3422:
3421:
3412:
3410:
3408:
3397:
3387:
3386:
3378:
3373:
3371:
3363:
3334:
3332:
3331:
3326:
3311:
3309:
3308:
3303:
3291:
3289:
3288:
3283:
3271:
3269:
3268:
3263:
3220:
3218:
3217:
3212:
3200:
3198:
3197:
3192:
3185:
3171:
3166:
3150:
3145:
3123:
3121:
3120:
3115:
3113:
3094:
3093:
3072:
3071:
3041:
3040:
3021:
3020:
2983:
2982:
2966:
2965:
2935:
2928:
2918:
2916:
2915:
2910:
2908:
2904:
2902:
2893:
2892:
2889:
2884:
2868:
2863:
2854:
2849:
2848:
2838:
2837:
2821:
2819:
2815:
2814:
2809:
2808:
2799:
2786:
2784:
2773:
2763:
2762:
2754:
2749:
2747:
2739:
2708:
2706:
2705:
2700:
2698:
2694:
2692:
2691:
2690:
2677:
2673:
2672:
2642:
2641:
2631:
2615:
2613:
2612:
2611:
2598:
2597:
2596:
2571:
2542:
2538:
2531:
2529:
2528:
2523:
2521:
2517:
2515:
2507:
2493:
2492:
2477:
2475:
2467:
2456:
2454:
2443:
2429:
2428:
2427:
2408:
2406:
2398:
2384:
2383:
2368:
2366:
2358:
2347:
2345:
2334:
2320:
2319:
2318:
2276:Ferrari's method
2273:
2266:
2264:
2263:
2258:
2234:
2233:
2218:
2217:
2202:
2201:
2179:
2170:
2161:
2152:
2139:
2137:
2136:
2131:
2108:
2107:
2092:
2091:
2076:
2075:
2047:
2045:
2044:
2039:
2034:
2033:
2012:
2011:
2002:
2001:
1982:
1978:
1974:
1966:
1962:
1958:
1943:
1942:
1940:
1939:
1933:
1930:
1919:
1908:
1904:
1897:
1890:
1883:
1879:
1868:
1864:
1860:
1856:
1849:
1841:
1837:
1826:
1817:
1813:
1806:
1802:
1795:
1784:
1770:
1768:
1767:
1762:
1760:
1759:
1738:
1737:
1719:
1718:
1700:
1699:
1690:
1689:
1671:
1670:
1639:
1637:
1636:
1631:
1602:
1601:
1589:
1588:
1569:
1568:
1566:
1565:
1559:
1556:
1543:
1541:
1540:
1535:
1515:
1514:
1496:
1495:
1466:
1465:
1463:
1462:
1456:
1453:
1440:
1438:
1437:
1432:
1430:
1429:
1386:
1384:
1383:
1378:
1376:
1372:
1371:
1362:
1361:
1352:
1351:
1336:
1335:
1326:
1325:
1310:
1309:
1300:
1299:
1275:
1274:
1259:
1258:
1249:
1248:
1233:
1232:
1223:
1222:
1201:
1194:
1193:
1175:
1174:
1144:
1143:
1119:
1118:
1109:
1108:
1090:
1089:
1077:
1076:
1055:
1051:
1050:
1041:
1040:
1022:
1021:
1009:
1008:
993:
992:
983:
982:
973:
972:
957:
956:
941:
940:
925:
924:
915:
914:
900:
875:
865:
863:
862:
857:
834:
833:
818:
817:
802:
801:
762:
760:
759:
754:
746:
744:the golden ratio
743:
731:
726:
725:
720:
711:
706:
704:
696:
688:
672:
668:
664:
660:
656:
649:
642:be the distinct
641:
637:
560:
481:Gerolamo Cardano
473:Lodovico Ferrari
429:
427:
426:
421:
410:
409:
394:
393:
349:, but, usually,
329:
319:
317:
316:
311:
285:
284:
269:
268:
253:
252:
228:quartic equation
202:
200:
199:
194:
174:
173:
158:
157:
142:
141:
99:quartic function
14404:
14403:
14399:
14398:
14397:
14395:
14394:
14393:
14369:
14368:
14367:
14362:
14311:
14250:
14193:Linear equation
14163:
14154:
14107:
14058:10.2307/2688990
14038:
14036:Further reading
14033:
14032:
14010:10.2307/2975214
13992:
13988:
13974:
13964:Springer-Verlag
13954:Euler, Leonhard
13951:
13947:
13933:
13923:Springer-Verlag
13910:
13906:
13872:
13866:
13862:
13848:
13828:Descartes, René
13825:
13821:
13792:
13788:
13773:10.2307/2972804
13757:
13753:
13739:10.2307/2305030
13721:
13717:
13710:
13694:
13690:
13676:10.2307/2589403
13655:
13651:
13619:
13615:
13591:
13587:
13577:
13575:
13565:
13561:
13554:math.gatech.edu
13549:
13545:
13544:
13540:
13531:
13527:
13521:
13500:
13496:
13472:
13468:
13458:
13456:
13446:
13442:
13437:
13408:
13381:Linear function
13377:
13366:
13360:
13356:
13353:
13346:
13340:
13334:
13327:
13320:
13314:
13311:
13304:
13297:
13291:
13288:
13281:
13274:
13268:
13264:
13245:
13232:
13231:
13230:
13227:
13224:
13223:
13214:
13208:
13205:
13198:
13192:
13175:
13174:
13168:
13164:
13148:
13121:
13117:
13114:
13113:
13104:
13100:
13064:
13060:
13053:
13026:
13022:
13018:
13016:
13013:
13012:
13006:quadratic forms
12991:
12984:
12979:
12974:
12967:
12960:
12955:
12941:
12930:
12911:
12907:
12888:
12878:
12856:
12818:
12814:
12802:
12798:
12780:
12776:
12758:
12754:
12752:
12749:
12748:
12728:
12727:
12717:
12712:
12698:
12694:
12692:
12679:
12661:
12657:
12650:
12635:
12631:
12628:
12627:
12617:
12612:
12598:
12594:
12592:
12579:
12561:
12557:
12550:
12535:
12531:
12527:
12525:
12522:
12521:
12504:
12493:
12488:
12484:
12479:
12475:
12470:
12460:
12427:
12423:
12414:
12410:
12392:
12388:
12376:
12372:
12357:
12353:
12351:
12348:
12347:
12335:
12327:
12322:
12312:
12308:
12303:
12299:
12294:
12268:
12261:
12257:
12256:
12255:
12246:
12242:
12230:
12223:
12219:
12218:
12217:
12208:
12204:
12192:
12185:
12181:
12180:
12179:
12170:
12166:
12161:
12158:
12157:
12151:
12145:
12142:
12136:
12133:
12127:
12119:
12116:
12111:
12110:
12108:
12106:
12100:
12096:
12091:
12070:
12069:
12057:
12053:
12044:
12040:
12031:
12027:
12018:
12014:
11999:
11992:
11986:
11982:
11979:
11978:
11966:
11962:
11953:
11949:
11940:
11936:
11927:
11923:
11908:
11901:
11895:
11891:
11888:
11887:
11875:
11871:
11862:
11858:
11849:
11845:
11836:
11832:
11817:
11810:
11804:
11800:
11797:
11796:
11784:
11780:
11771:
11767:
11758:
11754:
11745:
11741:
11726:
11719:
11713:
11709:
11705:
11703:
11700:
11699:
11674:
11670:
11666:
11660:
11656:
11651:
11644:Hadamard matrix
11638:resolvent cubic
11637:
11636:
11632:normal subgroup
11624:
11618:
11616:symmetric group
11612:
11596:
11587:is a root of (
11584:
11580:
11574:
11568:
11543:
11536:
11535:
11530:
11525:
11522:
11521:
11513:
11511:
11510:
11499:
11491:
11489:
11488:
11478:
11470:
11468:
11467:
11453:
11446:
11443:
11436:
11433:
11426:
11423:
11416:
11409:
11402:
11396:
11390:
11383:
11377:
11371:
11364:
11358:
11352:
11345:
11339:
11333:
11321:
11319:
11313:
11311:
11305:
11303:
11302:
11287:
11285:
11279:
11277:
11271:
11269:
11268:
11262:
11248:
11242:
11236:
11228:
11226:
11220:
11218:
11212:
11210:
11209:
11206:
11200:
11197:
11191:
11188:
11182:
11179:
11173:
11167:
11161:
11155:
11149:
11142:
11135:
11128:
11121:
11118:
11111:
11104:
11097:
11090:
11086:
11083:
11076:
11069:
11062:
11055:
11051:
11031:
11030:
11025:
11012:
11002:
10992:
10988:
10986:
10977:
10973:
10970:
10969:
10956:
10946:
10936:
10932:
10930:
10921:
10917:
10914:
10913:
10900:
10890:
10880:
10879:
10877:
10868:
10864:
10861:
10860:
10847:
10837:
10827:
10826:
10824:
10815:
10811:
10807:
10803:
10801:
10798:
10797:
10774:
10773:
10768:
10761:
10752:
10748:
10739:
10735:
10732:
10731:
10724:
10715:
10711:
10702:
10698:
10695:
10694:
10687:
10678:
10674:
10665:
10661:
10658:
10657:
10645:
10641:
10632:
10628:
10619:
10615:
10606:
10602:
10598:
10594:
10592:
10589:
10588:
10582:
10576:
10573:
10567:
10564:
10558:
10555:
10549:
10539:
10536:
10529:
10523:
10515:
10512:
10505:
10499:
10491:
10488:
10481:
10475:
10467:
10464:
10457:
10451:
10442:
10439:
10432:
10426:
10420:
10417:
10410:
10404:
10384:
10383:
10378:
10360:
10356:
10347:
10343:
10331:
10327:
10318:
10314:
10308:
10307:
10289:
10285:
10276:
10272:
10260:
10256:
10247:
10243:
10237:
10236:
10218:
10214:
10205:
10201:
10189:
10185:
10176:
10172:
10166:
10165:
10153:
10149:
10140:
10136:
10127:
10123:
10114:
10110:
10106:
10102:
10100:
10097:
10096:
10090:
10084:
10081:
10075:
10072:
10066:
10063:
10057:
10051:
10045:
10039:
10028:
10021:
10014:
10007:
10001:
9997:
9990:
9983:
9976:
9970:
9962:
9955:
9948:
9941:
9934:
9930:
9923:
9916:
9909:
9902:
9898:
9891:
9884:
9877:
9870:
9858:
9851:
9844:
9837:
9830:
9823:
9816:
9809:
9802:
9795:
9784:
9777:
9771:
9762:
9755:
9749:
9744:
9738:
9735:
9729:
9726:
9720:
9717:
9711:
9693:
9671:
9668:
9662:
9659:
9653:
9637:
9634:
9628:
9625:
9619:
9574:
9553:
9542:
9526:
9516:
9500:
9496:
9475:
9474:
9466:
9454:
9450:
9432:
9431:
9423:
9411:
9407:
9389:
9388:
9372:
9368:
9366:
9363:
9362:
9353:
9347:
9304:
9300:
9276:
9272:
9260:
9256:
9241:
9237:
9235:
9232:
9231:
9224:resolvent cubic
9213:
9196:
9195:
9186:
9182:
9170:
9169:
9157:
9153:
9141:
9140:
9110:
9106:
9097:
9096:
9018:
9014:
9005:
9004:
8998:
8994:
8976:
8972:
8963:
8959:
8941:
8937:
8930:
8924:
8920:
8911:
8907:
8901:
8897:
8882:
8878:
8874:
8872:
8869:
8868:
8861:
8857:
8837:
8836:
8821:
8820:
8793:
8792:
8774:
8770:
8760:
8756:
8754:
8751:
8750:
8737:
8733:
8727:
8721:
8710:
8692:
8669:
8668:
8653:
8652:
8628:
8627:
8600:
8599:
8580:
8576:
8574:
8571:
8570:
8546:
8545:
8503:
8499:
8466:
8462:
8438:
8434:
8425:
8424:
8400:
8396:
8369:
8365:
8355:
8334:
8330:
8318:
8314:
8302:
8298:
8294:
8292:
8289:
8288:
8282:
8264:
8262:
8261:
8258:general formula
8218:
8207:
8206:
8204:
8198:
8194:
8178:
8174:
8169:
8163:
8159:
8149:
8143:
8139:
8138:
8136:
8124:
8120:
8116:
8110:
8106:
8104:
8093:
8090:
8089:
8080:
8076:
8072:
8068:
8065:
8061:
8058:
8054:
8015:
8004:
8003:
8001:
7995:
7991:
7975:
7971:
7966:
7960:
7956:
7946:
7940:
7936:
7935:
7933:
7925:
7922:
7921:
7876:
7872:
7867:
7851:
7832:
7823:
7819:
7818:
7814:
7796:
7792:
7787:
7771:
7752:
7743:
7739:
7738:
7734:
7732:
7729:
7728:
7697:
7686:
7676:
7653:
7635:
7631:
7626:
7613:
7609:
7605:
7604:
7595:
7574:
7565:
7561:
7560:
7556:
7555:
7553:
7550:
7549:
7531:
7527:
7517:
7510:
7503:
7502:. This implies
7496:
7473:
7455:
7451:
7446:
7430:
7429:
7425:
7424:
7422:
7419:
7418:
7403:
7395:
7392:resolvent cubic
7352:
7348:
7324:
7320:
7305:
7301:
7286:
7282:
7277:
7274:
7273:
7222:
7218:
7216:
7198:
7194:
7193:
7189:
7165:
7161:
7150:
7147:
7146:
7139:
7131:
7119:
7112:
7065:
7061:
7059:
7041:
7037:
7019:
7015:
7000:
6979:
6970:
6966:
6965:
6961:
6960:
6958:
6955:
6954:
6945:
6927:
6923:
6895:
6891:
6889:
6862:
6847:
6838:
6834:
6833:
6829:
6828:
6826:
6823:
6822:
6774:
6770:
6758:
6754:
6752:
6749:
6748:
6739:
6726:
6719:
6716:
6714:
6708:
6707:
6705:
6703:
6697:
6689:
6686:
6681:
6680:
6678:
6676:
6670:
6667:
6661:
6644:
6643:
6631:
6624:
6620:
6619:
6618:
6614:
6607:
6603:
6597:
6590:
6586:
6585:
6584:
6578:
6574:
6562:
6555:
6551:
6550:
6549:
6543:
6539:
6533:
6529:
6517:
6510:
6506:
6505:
6504:
6498:
6494:
6482:
6475:
6471:
6470:
6469:
6462:
6460:
6442:
6438:
6405:
6401:
6394:
6392:
6385:
6379:
6378:
6369:
6362:
6358:
6357:
6356:
6352:
6345:
6338:
6334:
6333:
6332:
6326:
6322:
6310:
6306:
6300:
6296:
6290:
6286:
6274:
6267:
6263:
6262:
6261:
6260:
6258:
6222:
6218:
6217:
6215:
6208:
6202:
6201:
6192:
6185:
6181:
6180:
6179:
6175:
6168:
6161:
6157:
6156:
6155:
6143:
6139:
6133:
6129:
6125:
6123:
6108:
6104:
6091:
6089:
6082:
6075:
6073:
6070:
6069:
6047:
6043:
6035:
6032:
6027:
6026:
6024:
6019:
6018:. Substituting
6012:
6006:
6003:
6001:
5995:
5994:
5992:
5987:
5980:
5974:
5971:
5969:
5963:
5962:
5960:
5955:
5948:
5942:
5939:
5937:
5931:
5930:
5928:
5923:
5916:
5910:
5907:
5905:
5899:
5898:
5896:
5891:
5868:
5865:
5859:
5830:
5826:
5814:
5810:
5801:
5797:
5791:
5787:
5778:
5774:
5768:
5764:
5755:
5751:
5745:
5741:
5739:
5736:
5735:
5721:
5716:
5696:
5681:
5677:
5670:
5660:
5650:
5644:
5630:
5627:
5617:
5616:
5614:
5613:
5603:
5600:
5595:
5594:
5592:
5583:
5576:
5565:
5562:
5557:
5556:
5554:
5542:
5539:
5534:
5533:
5531:
5522:
5498:
5494:
5488:
5484:
5469:
5465:
5456:
5452:
5446:
5442:
5433:
5429:
5423:
5419:
5410:
5406:
5400:
5396:
5379:
5376:
5375:
5371:The polynomial
5369:
5350:
5349:
5338:
5334:
5332:
5322:
5316:
5312:
5309:
5308:
5297:
5293:
5291:
5281:
5275:
5271:
5268:
5267:
5256:
5252:
5250:
5240:
5234:
5230:
5227:
5226:
5215:
5211:
5209:
5199:
5193:
5189:
5185:
5183:
5180:
5179:
5165:
5154:
5151:
5145:
5142:
5136:
5133:
5123:
5113:
5099:
5093:
5087:
5073:
5063:
5032:
5028:
5019:
5015:
5009:
5005:
4996:
4992:
4986:
4982:
4965:
4962:
4961:
4954:
4947:
4941:
4938:
4906:
4900:
4887:
4871:
4847:
4821:
4817:
4805:
4801:
4792:
4788:
4782:
4778:
4766:
4762:
4750:
4746:
4737:
4733:
4727:
4723:
4703:
4700:
4699:
4673:
4669:
4657:
4653:
4644:
4640:
4634:
4630:
4621:
4617:
4611:
4607:
4595:
4591:
4565:
4562:
4561:
4547:
4528:
4517:
4490:
4463:
4459:
4447:
4443:
4434:
4430:
4424:
4420:
4411:
4407:
4401:
4397:
4388:
4384:
4378:
4374:
4357:
4354:
4353:
4347:
4342:
4306:
4302:
4300:
4297:
4296:
4274:
4271:
4270:
4246:
4231:
4227:
4215:
4211:
4209:
4206:
4205:
4188:
4184:
4182:
4179:
4178:
4133:
4129:
4115:
4112:
4111:
4094:
4090:
4088:
4085:
4084:
4054:
4050:
4048:
4045:
4044:
4018:
4014:
4012:
4009:
4008:
3986:
3983:
3982:
3955:
3952:
3951:
3927:
3911:
3905:
3898:
3894:
3893:
3891:
3888:
3887:
3865:
3862:
3861:
3845:
3842:
3841:
3816:
3813:
3812:
3790:
3786:
3784:
3781:
3780:
3760:
3756:
3754:
3751:
3750:
3731:
3726:
3720:
3718:
3715:
3714:
3689:
3686:
3685:
3666:
3661:
3655:
3644:
3639:
3623:
3618:
3612:
3610:
3607:
3606:
3580:
3576:
3574:
3571:
3570:
3548:
3545:
3544:
3509:
3504:
3498:
3494:
3488:
3484:
3482:
3478:
3464:
3461:
3460:
3431:
3417:
3413:
3411:
3401:
3396:
3377:
3372:
3362:
3354:
3351:
3350:
3317:
3314:
3313:
3297:
3294:
3293:
3277:
3274:
3273:
3248:
3245:
3244:
3240:
3206:
3203:
3202:
3167:
3162:
3146:
3141:
3135:
3132:
3131:
3111:
3110:
3089:
3085:
3067:
3063:
3036:
3032:
3022:
3016:
3012:
3009:
3008:
2978:
2974:
2967:
2961:
2957:
2953:
2951:
2948:
2947:
2930:
2923:
2906:
2905:
2898:
2885:
2880:
2864:
2859:
2853:
2844:
2840:
2839:
2836:
2829:
2823:
2822:
2804:
2800:
2798:
2791:
2787:
2777:
2772:
2753:
2748:
2738:
2731:
2724:
2722:
2719:
2718:
2696:
2695:
2686:
2682:
2678:
2668:
2664:
2637:
2633:
2632:
2630:
2623:
2617:
2616:
2607:
2603:
2599:
2592:
2588:
2572:
2570:
2563:
2556:
2554:
2551:
2550:
2540:
2536:
2519:
2518:
2506:
2488:
2484:
2476:
2466:
2447:
2442:
2432:
2417:
2413:
2410:
2409:
2397:
2379:
2375:
2367:
2357:
2338:
2333:
2323:
2308:
2304:
2300:
2298:
2295:
2294:
2288:cubic equations
2271:
2229:
2225:
2213:
2209:
2197:
2193:
2188:
2185:
2184:
2178:
2172:
2169:
2163:
2160:
2154:
2151:
2145:
2144:The four roots
2103:
2099:
2087:
2083:
2071:
2067:
2065:
2062:
2061:
2054:
2029:
2025:
2007:
2003:
1997:
1993:
1988:
1985:
1984:
1980:
1976:
1972:
1968:
1964:
1960:
1956:
1952:
1934:
1931:
1926:
1925:
1923:
1921:
1917:
1913:
1906:
1902:
1895:
1888:
1881:
1877:
1873:
1866:
1862:
1858:
1854:
1847:
1843:
1839:
1835:
1824:
1815:
1811:
1804:
1800:
1793:
1789:non-real roots.
1782:
1755:
1751:
1733:
1729:
1714:
1710:
1695:
1691:
1685:
1681:
1666:
1662:
1651:
1648:
1647:
1597:
1593:
1584:
1580:
1578:
1575:
1574:
1560:
1557:
1552:
1551:
1549:
1548:
1510:
1506:
1491:
1487:
1479:
1476:
1475:
1457:
1454:
1449:
1448:
1446:
1445:
1425:
1421:
1398:
1395:
1394:
1374:
1373:
1367:
1363:
1357:
1353:
1347:
1343:
1331:
1327:
1321:
1317:
1305:
1301:
1295:
1291:
1270:
1266:
1254:
1250:
1244:
1240:
1228:
1224:
1218:
1214:
1199:
1198:
1189:
1185:
1170:
1166:
1139:
1135:
1114:
1110:
1104:
1100:
1085:
1081:
1072:
1068:
1053:
1052:
1046:
1042:
1036:
1032:
1017:
1013:
1004:
1000:
988:
984:
978:
974:
968:
964:
952:
948:
936:
932:
920:
916:
910:
906:
901:
899:
889:
887:
884:
883:
870:
829:
825:
813:
809:
797:
793:
788:
785:
784:
778:
773:
742:
719:
712:
710:
697:
689:
687:
685:
682:
681:
670:
666:
662:
658:
654:
647:
639:
635:
632:
601:of the matrix.
556:
510:
502:complete theory
498:Évariste Galois
470:
445:. Likewise, if
405:
401:
389:
385:
365:
362:
361:
324:
280:
276:
264:
260:
248:
244:
239:
236:
235:
218:four, called a
169:
165:
153:
149:
137:
133:
113:
110:
109:
69:were above the
48:critical points
38:
31:
24:
17:
12:
11:
5:
14402:
14392:
14391:
14386:
14381:
14364:
14363:
14361:
14360:
14355:
14350:
14345:
14340:
14335:
14330:
14325:
14319:
14317:
14313:
14312:
14310:
14309:
14304:
14299:
14294:
14289:
14284:
14279:
14274:
14269:
14264:
14258:
14256:
14252:
14251:
14249:
14248:
14243:
14238:
14233:
14232:
14231:
14221:
14220:
14219:
14217:Cubic equation
14209:
14208:
14207:
14197:
14196:
14195:
14185:
14180:
14174:
14172:
14165:
14164:
14153:
14152:
14145:
14138:
14130:
14124:
14123:
14118:
14106:
14105:External links
14103:
14102:
14101:
14070:
14037:
14034:
14031:
14030:
13986:
13972:
13945:
13931:
13904:
13860:
13846:
13819:
13786:
13751:
13733:(3): 165–170,
13715:
13708:
13688:
13670:(6): 523–528,
13649:
13613:
13585:
13559:
13538:
13532:Stewart, Ian,
13525:
13519:
13494:
13466:
13439:
13438:
13436:
13433:
13432:
13431:
13428:Euler's number
13407:
13404:
13403:
13402:
13396:
13393:Cubic function
13390:
13384:
13376:
13373:
13351:
13344:
13332:
13325:
13318:
13309:
13302:
13295:
13286:
13279:
13272:
13248:
13243:
13240:
13235:
13203:
13196:
13189:
13188:
13171:
13167:
13163:
13160:
13157:
13154:
13151:
13149:
13147:
13144:
13141:
13138:
13135:
13132:
13129:
13124:
13120:
13116:
13115:
13112:
13107:
13103:
13099:
13096:
13093:
13090:
13087:
13084:
13081:
13078:
13075:
13072:
13067:
13063:
13059:
13056:
13054:
13052:
13049:
13046:
13043:
13040:
13037:
13034:
13029:
13025:
13021:
13020:
12982:
12972:
12965:
12958:
12877:
12874:
12853:
12852:
12841:
12838:
12835:
12832:
12829:
12826:
12821:
12817:
12813:
12810:
12805:
12801:
12797:
12794:
12791:
12788:
12783:
12779:
12775:
12772:
12769:
12766:
12761:
12757:
12742:
12741:
12723:
12720:
12716:
12711:
12706:
12701:
12697:
12691:
12686:
12683:
12678:
12675:
12672:
12669:
12664:
12660:
12656:
12653:
12651:
12649:
12646:
12643:
12638:
12634:
12630:
12629:
12623:
12620:
12616:
12611:
12606:
12601:
12597:
12591:
12586:
12583:
12578:
12575:
12572:
12569:
12564:
12560:
12556:
12553:
12551:
12549:
12546:
12543:
12538:
12534:
12530:
12529:
12491:
12482:
12473:
12467:cubic function
12455:
12454:
12445:
12443:
12430:
12426:
12422:
12417:
12413:
12409:
12406:
12403:
12400:
12395:
12391:
12387:
12384:
12379:
12375:
12371:
12368:
12365:
12360:
12356:
12325:
12306:
12297:
12291:
12290:
12279:
12276:
12271:
12264:
12260:
12254:
12249:
12245:
12241:
12238:
12233:
12226:
12222:
12216:
12211:
12207:
12203:
12200:
12195:
12188:
12184:
12178:
12173:
12169:
12165:
12149:
12140:
12131:
12104:
12094:
12084:
12083:
12068:
12065:
12060:
12056:
12052:
12047:
12043:
12039:
12034:
12030:
12026:
12021:
12017:
12013:
12007:
12004:
11998:
11995:
11993:
11989:
11985:
11981:
11980:
11977:
11974:
11969:
11965:
11961:
11956:
11952:
11948:
11943:
11939:
11935:
11930:
11926:
11922:
11916:
11913:
11907:
11904:
11902:
11898:
11894:
11890:
11889:
11886:
11883:
11878:
11874:
11870:
11865:
11861:
11857:
11852:
11848:
11844:
11839:
11835:
11831:
11825:
11822:
11816:
11813:
11811:
11807:
11803:
11799:
11798:
11795:
11792:
11787:
11783:
11779:
11774:
11770:
11766:
11761:
11757:
11753:
11748:
11744:
11740:
11734:
11731:
11725:
11722:
11720:
11716:
11712:
11708:
11707:
11654:
11622:
11611:
11608:
11565:
11564:
11547:
11540:
11534:
11529:
11505:
11451:
11441:
11431:
11421:
11406:
11405:
11400:
11394:
11388:
11381:
11375:
11369:
11362:
11356:
11350:
11343:
11337:
11331:
11204:
11195:
11186:
11177:
11147:
11140:
11133:
11126:
11116:
11109:
11102:
11095:
11081:
11074:
11067:
11060:
11048:
11047:
11035:
11022:
11016:
11011:
11006:
11001:
10996:
10991:
10985:
10980:
10976:
10972:
10971:
10966:
10960:
10955:
10950:
10945:
10940:
10935:
10929:
10924:
10920:
10916:
10915:
10910:
10904:
10899:
10894:
10889:
10884:
10876:
10871:
10867:
10863:
10862:
10857:
10851:
10846:
10841:
10836:
10831:
10823:
10818:
10814:
10810:
10809:
10806:
10791:
10790:
10778:
10765:
10760:
10755:
10751:
10747:
10742:
10738:
10734:
10733:
10728:
10723:
10718:
10714:
10710:
10705:
10701:
10697:
10696:
10691:
10686:
10681:
10677:
10673:
10668:
10664:
10660:
10659:
10656:
10653:
10648:
10644:
10640:
10635:
10631:
10627:
10622:
10618:
10614:
10609:
10605:
10601:
10600:
10597:
10584:are such that
10580:
10571:
10562:
10553:
10546:
10545:
10534:
10527:
10521:
10510:
10503:
10497:
10486:
10479:
10473:
10462:
10455:
10437:
10430:
10415:
10408:
10401:
10400:
10388:
10377:
10374:
10371:
10368:
10363:
10359:
10355:
10350:
10346:
10342:
10339:
10334:
10330:
10326:
10321:
10317:
10313:
10310:
10309:
10306:
10303:
10300:
10297:
10292:
10288:
10284:
10279:
10275:
10271:
10268:
10263:
10259:
10255:
10250:
10246:
10242:
10239:
10238:
10235:
10232:
10229:
10226:
10221:
10217:
10213:
10208:
10204:
10200:
10197:
10192:
10188:
10184:
10179:
10175:
10171:
10168:
10167:
10164:
10161:
10156:
10152:
10148:
10143:
10139:
10135:
10130:
10126:
10122:
10117:
10113:
10109:
10108:
10105:
10092:are such that
10088:
10079:
10070:
10061:
10026:
10019:
10012:
10005:
9995:
9988:
9981:
9974:
9960:
9953:
9946:
9939:
9928:
9921:
9914:
9907:
9896:
9889:
9882:
9875:
9856:
9849:
9842:
9835:
9821:
9814:
9807:
9800:
9792:
9791:
9782:
9775:
9769:
9760:
9753:
9747:
9742:
9733:
9724:
9715:
9686:
9685:
9666:
9657:
9651:
9632:
9623:
9617:
9541:
9538:
9492:
9491:
9479:
9473:
9469:
9465:
9462:
9457:
9453:
9449:
9446:
9443:
9440:
9437:
9434:
9433:
9430:
9426:
9422:
9419:
9414:
9410:
9406:
9403:
9400:
9397:
9394:
9391:
9390:
9387:
9384:
9381:
9378:
9375:
9374:
9371:
9341:done elsewhere
9335:
9334:
9325:
9323:
9312:
9307:
9303:
9299:
9296:
9293:
9290:
9287:
9284:
9279:
9275:
9271:
9268:
9263:
9259:
9255:
9252:
9249:
9244:
9240:
9210:
9209:
9194:
9189:
9185:
9181:
9178:
9175:
9173:
9171:
9168:
9165:
9160:
9156:
9152:
9149:
9146:
9144:
9142:
9139:
9136:
9133:
9130:
9127:
9124:
9121:
9118:
9113:
9109:
9105:
9102:
9100:
9098:
9095:
9092:
9089:
9086:
9083:
9080:
9077:
9074:
9071:
9068:
9065:
9062:
9059:
9056:
9053:
9050:
9047:
9044:
9041:
9038:
9035:
9032:
9029:
9026:
9021:
9017:
9013:
9010:
9008:
9006:
9001:
8997:
8993:
8990:
8987:
8984:
8979:
8975:
8971:
8966:
8962:
8958:
8955:
8952:
8949:
8944:
8940:
8936:
8933:
8931:
8927:
8923:
8919:
8914:
8910:
8904:
8900:
8896:
8893:
8890:
8885:
8881:
8877:
8876:
8854:
8853:
8841:
8835:
8832:
8829:
8826:
8823:
8822:
8819:
8816:
8813:
8810:
8807:
8804:
8801:
8798:
8795:
8794:
8791:
8788:
8785:
8782:
8777:
8773:
8769:
8766:
8763:
8762:
8759:
8686:
8685:
8673:
8667:
8664:
8661:
8658:
8655:
8654:
8651:
8648:
8645:
8642:
8639:
8636:
8633:
8630:
8629:
8626:
8623:
8620:
8617:
8614:
8611:
8608:
8605:
8602:
8601:
8598:
8595:
8592:
8589:
8586:
8583:
8582:
8579:
8560:
8559:
8544:
8541:
8538:
8535:
8532:
8529:
8526:
8523:
8520:
8517:
8514:
8511:
8506:
8502:
8498:
8495:
8492:
8489:
8486:
8483:
8480:
8477:
8474:
8469:
8465:
8461:
8458:
8455:
8452:
8449:
8446:
8441:
8437:
8433:
8430:
8428:
8426:
8423:
8420:
8417:
8414:
8411:
8408:
8403:
8399:
8395:
8392:
8389:
8386:
8383:
8380:
8377:
8372:
8368:
8364:
8361:
8358:
8356:
8354:
8351:
8348:
8345:
8342:
8337:
8333:
8329:
8326:
8321:
8317:
8313:
8310:
8305:
8301:
8297:
8296:
8281:
8278:
8254:
8253:
8242:
8237:
8230:
8222:
8216:
8211:
8201:
8197:
8193:
8190:
8187:
8184:
8181:
8177:
8173:
8166:
8162:
8156:
8153:
8146:
8142:
8135:
8127:
8123:
8119:
8113:
8109:
8103:
8100:
8097:
8078:
8067:denote either
8063:
8056:
8051:
8050:
8039:
8034:
8027:
8019:
8013:
8008:
7998:
7994:
7990:
7987:
7984:
7981:
7978:
7974:
7970:
7963:
7959:
7953:
7950:
7943:
7939:
7932:
7929:
7911:
7910:
7899:
7896:
7892:
7883:
7880:
7875:
7871:
7866:
7863:
7858:
7855:
7850:
7847:
7844:
7839:
7836:
7831:
7826:
7822:
7817:
7812:
7803:
7800:
7795:
7791:
7786:
7783:
7778:
7775:
7770:
7767:
7764:
7759:
7756:
7751:
7746:
7742:
7737:
7673:
7672:
7661:
7656:
7651:
7642:
7639:
7634:
7630:
7625:
7620:
7617:
7612:
7608:
7603:
7598:
7593:
7589:
7586:
7581:
7578:
7573:
7568:
7564:
7559:
7493:
7492:
7481:
7476:
7471:
7462:
7459:
7454:
7450:
7445:
7442:
7437:
7434:
7428:
7386:
7385:
7376:
7374:
7363:
7360:
7355:
7351:
7347:
7344:
7341:
7338:
7335:
7332:
7327:
7323:
7319:
7316:
7313:
7308:
7304:
7300:
7297:
7294:
7289:
7285:
7281:
7265:
7264:
7252:
7249:
7246:
7242:
7238:
7235:
7230:
7225:
7221:
7215:
7212:
7209:
7206:
7201:
7197:
7192:
7188:
7185:
7182:
7179:
7176:
7173:
7168:
7164:
7160:
7157:
7154:
7107:
7106:
7097:
7095:
7084:
7081:
7078:
7073:
7068:
7064:
7058:
7055:
7052:
7049:
7044:
7040:
7036:
7033:
7030:
7027:
7022:
7018:
7014:
7011:
7008:
7003:
6998:
6994:
6991:
6986:
6983:
6978:
6973:
6969:
6964:
6920:
6919:
6908:
6903:
6898:
6894:
6888:
6885:
6882:
6879:
6876:
6873:
6870:
6865:
6860:
6854:
6851:
6846:
6841:
6837:
6832:
6812:
6811:
6800:
6797:
6794:
6791:
6788:
6785:
6782:
6777:
6773:
6769:
6766:
6761:
6757:
6738:
6735:
6724:
6712:
6701:
6674:
6665:
6658:
6657:
6642:
6634:
6627:
6623:
6617:
6610:
6606:
6600:
6593:
6589:
6581:
6577:
6573:
6570:
6565:
6558:
6554:
6546:
6542:
6536:
6532:
6528:
6525:
6520:
6513:
6509:
6501:
6497:
6493:
6490:
6485:
6478:
6474:
6468:
6465:
6459:
6454:
6450:
6445:
6441:
6437:
6434:
6431:
6428:
6425:
6422:
6419:
6416:
6413:
6408:
6404:
6400:
6397:
6391:
6388:
6386:
6384:
6381:
6380:
6372:
6365:
6361:
6355:
6348:
6341:
6337:
6329:
6325:
6321:
6318:
6313:
6309:
6303:
6299:
6293:
6289:
6285:
6282:
6277:
6270:
6266:
6257:
6252:
6248:
6245:
6242:
6239:
6236:
6233:
6230:
6225:
6221:
6214:
6211:
6209:
6207:
6204:
6203:
6195:
6188:
6184:
6178:
6171:
6164:
6160:
6154:
6151:
6146:
6142:
6136:
6132:
6128:
6122:
6117:
6111:
6107:
6103:
6100:
6097:
6094:
6088:
6085:
6083:
6081:
6078:
6077:
6010:
5999:
5978:
5967:
5946:
5935:
5914:
5903:
5863:
5853:
5852:
5841:
5838:
5833:
5829:
5825:
5822:
5817:
5813:
5809:
5804:
5800:
5794:
5790:
5786:
5781:
5777:
5771:
5767:
5763:
5758:
5754:
5748:
5744:
5720:
5717:
5715:
5712:
5675:
5668:
5658:
5648:
5515:
5514:
5501:
5497:
5491:
5487:
5483:
5480:
5477:
5472:
5468:
5464:
5459:
5455:
5449:
5445:
5441:
5436:
5432:
5426:
5422:
5418:
5413:
5409:
5403:
5399:
5395:
5392:
5389:
5386:
5383:
5368:
5365:
5364:
5363:
5348:
5341:
5337:
5331:
5328:
5325:
5323:
5319:
5315:
5311:
5310:
5307:
5300:
5296:
5290:
5287:
5284:
5282:
5278:
5274:
5270:
5269:
5266:
5259:
5255:
5249:
5246:
5243:
5241:
5237:
5233:
5229:
5228:
5225:
5218:
5214:
5208:
5205:
5202:
5200:
5196:
5192:
5188:
5187:
5149:
5140:
5131:
5121:
5111:
5049:
5048:
5035:
5031:
5027:
5022:
5018:
5012:
5008:
5004:
4999:
4995:
4989:
4985:
4981:
4978:
4975:
4972:
4969:
4952:
4945:
4937:
4934:
4930:Euler's method
4918:
4917:
4897:
4863:cubic function
4844:
4843:
4832:
4829:
4824:
4820:
4816:
4813:
4808:
4804:
4800:
4795:
4791:
4785:
4781:
4777:
4774:
4769:
4765:
4761:
4758:
4753:
4749:
4745:
4740:
4736:
4730:
4726:
4722:
4719:
4716:
4713:
4710:
4707:
4693:
4692:
4681:
4676:
4672:
4668:
4665:
4660:
4656:
4652:
4647:
4643:
4637:
4633:
4629:
4624:
4620:
4614:
4610:
4606:
4603:
4598:
4594:
4590:
4587:
4584:
4581:
4578:
4575:
4572:
4569:
4483:
4482:
4471:
4466:
4462:
4458:
4455:
4450:
4446:
4442:
4437:
4433:
4427:
4423:
4419:
4414:
4410:
4404:
4400:
4396:
4391:
4387:
4381:
4377:
4373:
4370:
4367:
4364:
4361:
4346:
4343:
4341:
4338:
4337:
4336:
4320:
4317:
4314:
4309:
4305:
4284:
4281:
4278:
4267:
4256:
4253:
4249:
4245:
4242:
4239:
4234:
4230:
4226:
4223:
4218:
4214:
4191:
4187:
4162:
4159:
4156:
4153:
4150:
4147:
4144:
4141:
4136:
4132:
4128:
4125:
4122:
4119:
4097:
4093:
4068:
4065:
4062:
4057:
4053:
4043:and thus also
4032:
4029:
4026:
4021:
4017:
3996:
3993:
3990:
3979:
3959:
3935:
3930:
3925:
3917:
3914:
3910:
3904:
3901:
3897:
3875:
3872:
3869:
3849:
3829:
3826:
3823:
3820:
3809:
3798:
3793:
3789:
3768:
3763:
3759:
3734:
3729:
3725:
3702:
3699:
3696:
3693:
3669:
3664:
3660:
3654:
3647:
3642:
3638:
3634:
3631:
3626:
3621:
3617:
3594:
3591:
3588:
3583:
3579:
3558:
3555:
3552:
3540:
3539:
3538:
3537:
3526:
3522:
3512:
3507:
3503:
3497:
3491:
3487:
3481:
3477:
3474:
3471:
3468:
3455:
3454:
3453:
3438:
3435:
3430:
3427:
3420:
3416:
3407:
3404:
3400:
3395:
3392:
3384:
3381:
3376:
3369:
3366:
3361:
3358:
3345:
3344:
3324:
3321:
3301:
3281:
3261:
3258:
3255:
3252:
3239:
3236:
3235:
3234:
3210:
3190:
3184:
3181:
3178:
3175:
3170:
3165:
3161:
3157:
3154:
3149:
3144:
3140:
3125:
3124:
3109:
3106:
3103:
3100:
3097:
3092:
3088:
3084:
3081:
3078:
3075:
3070:
3066:
3062:
3059:
3056:
3053:
3050:
3047:
3044:
3039:
3035:
3031:
3028:
3025:
3023:
3019:
3015:
3011:
3010:
3007:
3004:
3001:
2998:
2995:
2992:
2989:
2986:
2981:
2977:
2973:
2970:
2968:
2964:
2960:
2956:
2955:
2920:
2919:
2901:
2896:
2888:
2883:
2879:
2875:
2872:
2867:
2862:
2858:
2852:
2847:
2843:
2835:
2832:
2830:
2828:
2825:
2824:
2818:
2812:
2807:
2803:
2797:
2794:
2790:
2783:
2780:
2776:
2771:
2768:
2760:
2757:
2752:
2745:
2742:
2737:
2734:
2732:
2730:
2727:
2726:
2712:
2711:
2709:
2689:
2685:
2681:
2676:
2671:
2667:
2663:
2660:
2657:
2654:
2651:
2648:
2645:
2640:
2636:
2629:
2626:
2624:
2622:
2619:
2618:
2610:
2606:
2602:
2595:
2591:
2587:
2584:
2581:
2578:
2575:
2569:
2566:
2564:
2562:
2559:
2558:
2533:
2532:
2513:
2510:
2505:
2502:
2499:
2496:
2491:
2487:
2483:
2480:
2473:
2470:
2465:
2462:
2459:
2453:
2450:
2446:
2441:
2438:
2435:
2433:
2426:
2423:
2420:
2416:
2412:
2411:
2404:
2401:
2396:
2393:
2390:
2387:
2382:
2378:
2374:
2371:
2364:
2361:
2356:
2353:
2350:
2344:
2341:
2337:
2332:
2329:
2326:
2324:
2317:
2314:
2311:
2307:
2303:
2302:
2268:
2267:
2255:
2252:
2249:
2246:
2243:
2240:
2237:
2232:
2228:
2224:
2221:
2216:
2212:
2208:
2205:
2200:
2196:
2192:
2176:
2167:
2158:
2149:
2129:
2126:
2123:
2120:
2117:
2114:
2111:
2106:
2102:
2098:
2095:
2090:
2086:
2082:
2079:
2074:
2070:
2053:
2050:
2037:
2032:
2028:
2024:
2021:
2018:
2015:
2010:
2006:
2000:
1996:
1992:
1983:> 0, since
1970:
1954:
1949:
1948:
1947:
1946:
1945:
1944:
1915:
1910:
1899:
1885:
1875:
1870:
1851:
1845:
1821:
1820:
1819:
1808:
1790:
1772:
1771:
1758:
1754:
1750:
1747:
1744:
1741:
1736:
1732:
1728:
1725:
1722:
1717:
1713:
1709:
1706:
1703:
1698:
1694:
1688:
1684:
1680:
1677:
1674:
1669:
1665:
1661:
1658:
1655:
1641:
1640:
1629:
1626:
1623:
1620:
1617:
1614:
1611:
1608:
1605:
1600:
1596:
1592:
1587:
1583:
1545:
1544:
1533:
1530:
1527:
1524:
1521:
1518:
1513:
1509:
1505:
1502:
1499:
1494:
1490:
1486:
1483:
1442:
1441:
1428:
1424:
1420:
1417:
1414:
1411:
1408:
1405:
1402:
1388:
1387:
1370:
1366:
1360:
1356:
1350:
1346:
1342:
1339:
1334:
1330:
1324:
1320:
1316:
1313:
1308:
1304:
1298:
1294:
1290:
1287:
1284:
1281:
1278:
1273:
1269:
1265:
1262:
1257:
1253:
1247:
1243:
1239:
1236:
1231:
1227:
1221:
1217:
1213:
1210:
1207:
1204:
1202:
1200:
1197:
1192:
1188:
1184:
1181:
1178:
1173:
1169:
1165:
1162:
1159:
1156:
1153:
1150:
1147:
1142:
1138:
1134:
1131:
1128:
1125:
1122:
1117:
1113:
1107:
1103:
1099:
1096:
1093:
1088:
1084:
1080:
1075:
1071:
1067:
1064:
1061:
1058:
1056:
1054:
1049:
1045:
1039:
1035:
1031:
1028:
1025:
1020:
1016:
1012:
1007:
1003:
999:
996:
991:
987:
981:
977:
971:
967:
963:
960:
955:
951:
947:
944:
939:
935:
931:
928:
923:
919:
913:
909:
905:
902:
898:
895:
892:
891:
867:
866:
855:
852:
849:
846:
843:
840:
837:
832:
828:
824:
821:
816:
812:
808:
805:
800:
796:
792:
777:
774:
772:
769:
764:
763:
752:
749:
741:
737:
734:
729:
723:
718:
715:
709:
703:
700:
695:
692:
675:golden section
631:
628:
518:conic sections
509:
506:
469:
466:
443:global minimum
431:
430:
419:
416:
413:
408:
404:
400:
397:
392:
388:
384:
381:
378:
375:
372:
369:
336:cubic function
321:
320:
309:
306:
303:
300:
297:
294:
291:
288:
283:
279:
275:
272:
267:
263:
259:
256:
251:
247:
243:
204:
203:
192:
189:
186:
183:
180:
177:
172:
168:
164:
161:
156:
152:
148:
145:
140:
136:
132:
129:
126:
123:
120:
117:
15:
9:
6:
4:
3:
2:
14401:
14390:
14387:
14385:
14382:
14380:
14377:
14376:
14374:
14359:
14358:Gröbner basis
14356:
14354:
14351:
14349:
14346:
14344:
14341:
14339:
14336:
14334:
14331:
14329:
14326:
14324:
14323:Factorization
14321:
14320:
14318:
14314:
14308:
14305:
14303:
14300:
14298:
14295:
14293:
14290:
14288:
14285:
14283:
14280:
14278:
14275:
14273:
14270:
14268:
14265:
14263:
14260:
14259:
14257:
14255:By properties
14253:
14247:
14244:
14242:
14239:
14237:
14234:
14230:
14227:
14226:
14225:
14222:
14218:
14215:
14214:
14213:
14210:
14206:
14203:
14202:
14201:
14198:
14194:
14191:
14190:
14189:
14186:
14184:
14181:
14179:
14176:
14175:
14173:
14171:
14166:
14162:
14158:
14151:
14146:
14144:
14139:
14137:
14132:
14131:
14128:
14122:
14119:
14116:
14112:
14109:
14108:
14098:
14094:
14090:
14086:
14082:
14078:
14077:
14071:
14067:
14063:
14059:
14055:
14051:
14047:
14046:
14040:
14039:
14027:
14023:
14019:
14015:
14011:
14007:
14003:
13999:
13998:
13990:
13983:
13979:
13975:
13969:
13965:
13961:
13960:
13955:
13949:
13942:
13938:
13934:
13932:0-387-97424-5
13928:
13924:
13920:
13919:
13914:
13908:
13900:
13896:
13892:
13888:
13884:
13880:
13879:
13871:
13864:
13857:
13853:
13849:
13847:0-486-60068-8
13843:
13839:
13835:
13834:
13829:
13823:
13814:
13809:
13805:
13801:
13797:
13790:
13782:
13778:
13774:
13770:
13766:
13762:
13755:
13748:
13744:
13740:
13736:
13732:
13728:
13727:
13719:
13711:
13705:
13701:
13700:
13692:
13685:
13681:
13677:
13673:
13669:
13665:
13664:
13659:
13653:
13646:
13642:
13637:
13632:
13629:(3): 254–78,
13628:
13624:
13617:
13610:
13606:
13605:
13600:
13596:
13589:
13574:
13570:
13563:
13555:
13548:
13542:
13535:
13529:
13522:
13520:0-486-67811-3
13516:
13511:
13510:
13504:
13498:
13491:
13487:
13486:
13481:
13477:
13470:
13455:
13451:
13444:
13440:
13429:
13425:
13422:is used as a
13421:
13416:
13415:
13410:
13409:
13400:
13397:
13394:
13391:
13388:
13385:
13382:
13379:
13378:
13372:
13369:
13363:
13350:
13343:
13337:
13331:
13324:
13317:
13308:
13301:
13294:
13285:
13278:
13271:
13263: =
13241:
13238:
13220:
13217:
13211:
13202:
13195:
13169:
13165:
13161:
13158:
13155:
13152:
13150:
13142:
13139:
13136:
13133:
13130:
13122:
13118:
13110:
13105:
13101:
13097:
13094:
13091:
13088:
13085:
13082:
13079:
13076:
13073:
13070:
13065:
13061:
13057:
13055:
13047:
13044:
13041:
13038:
13035:
13027:
13023:
13011:
13010:
13009:
13007:
13003:
12998:
12994:
12988:
12985:
12975:
12968:
12961:
12953:
12948:
12944:
12937:
12933:
12926:
12922:
12918:
12914:
12903:
12899:
12895:
12891:
12885:
12883:
12873:
12871:
12866:
12864:
12859:
12839:
12836:
12833:
12830:
12827:
12824:
12819:
12815:
12811:
12808:
12803:
12799:
12795:
12789:
12781:
12777:
12773:
12767:
12759:
12755:
12747:
12746:
12745:
12721:
12718:
12714:
12709:
12704:
12699:
12695:
12689:
12684:
12681:
12676:
12673:
12670:
12667:
12662:
12658:
12654:
12652:
12644:
12636:
12632:
12621:
12618:
12614:
12609:
12604:
12599:
12595:
12589:
12584:
12581:
12576:
12573:
12570:
12567:
12562:
12558:
12554:
12552:
12544:
12536:
12532:
12520:
12519:
12518:
12516:
12515:
12514:
12507:
12502:
12497:
12494:
12485:
12476:
12468:
12463:
12453:
12446:
12444:
12428:
12424:
12420:
12415:
12411:
12404:
12401:
12398:
12393:
12389:
12382:
12377:
12373:
12369:
12366:
12363:
12358:
12354:
12346:
12345:
12342:
12338:
12333:
12328:
12320:
12315:
12309:
12300:
12277:
12269:
12262:
12258:
12252:
12247:
12243:
12231:
12224:
12220:
12214:
12209:
12205:
12193:
12186:
12182:
12176:
12171:
12167:
12156:
12155:
12154:
12148:
12139:
12130:
12114:
12103:
12097:
12089:
12066:
12058:
12054:
12050:
12045:
12041:
12037:
12032:
12028:
12024:
12019:
12015:
12005:
12002:
11996:
11994:
11987:
11983:
11975:
11967:
11963:
11959:
11954:
11950:
11946:
11941:
11937:
11933:
11928:
11924:
11914:
11911:
11905:
11903:
11896:
11892:
11884:
11876:
11872:
11868:
11863:
11859:
11855:
11850:
11846:
11842:
11837:
11833:
11823:
11820:
11814:
11812:
11805:
11801:
11793:
11785:
11781:
11777:
11772:
11768:
11764:
11759:
11755:
11751:
11746:
11742:
11732:
11729:
11723:
11721:
11714:
11710:
11698:
11697:
11696:
11693:
11689:
11685:
11681:
11677:
11663:
11657:
11649:
11645:
11641:
11633:
11629:
11621:
11617:
11607:
11605:
11599:
11594:
11593:
11592:
11577:
11571:
11545:
11538:
11532:
11527:
11516:
11509:
11506:
11502:
11494:
11486:
11481:
11473:
11465:
11461:
11460:
11459:
11456:
11450:
11440:
11430:
11420:
11413:
11399:
11393:
11387:
11380:
11374:
11368:
11361:
11355:
11349:
11342:
11336:
11330:
11324:
11316:
11308:
11301:
11300:
11299:
11296:
11290:
11282:
11274:
11265:
11260:
11259:
11258:
11251:
11245:
11239:
11231:
11223:
11215:
11203:
11194:
11185:
11176:
11170:
11164:
11158:
11152:
11146:
11139:
11132:
11125:
11115:
11108:
11101:
11094:
11080:
11073:
11066:
11059:
11020:
11014:
11009:
11004:
10999:
10994:
10989:
10983:
10978:
10974:
10964:
10958:
10953:
10948:
10943:
10938:
10933:
10927:
10922:
10918:
10908:
10902:
10897:
10892:
10887:
10882:
10874:
10869:
10865:
10855:
10849:
10844:
10839:
10834:
10829:
10821:
10816:
10812:
10804:
10796:
10795:
10794:
10763:
10758:
10753:
10749:
10745:
10740:
10736:
10726:
10721:
10716:
10712:
10708:
10703:
10699:
10689:
10684:
10679:
10675:
10671:
10666:
10662:
10654:
10651:
10646:
10642:
10638:
10633:
10629:
10625:
10620:
10616:
10612:
10607:
10603:
10595:
10587:
10586:
10585:
10579:
10570:
10561:
10552:
10542:
10533:
10526:
10522:
10518:
10509:
10502:
10498:
10494:
10485:
10478:
10474:
10470:
10461:
10454:
10450:
10449:
10448:
10445:
10436:
10429:
10423:
10414:
10407:
10375:
10372:
10369:
10361:
10357:
10353:
10348:
10344:
10332:
10328:
10324:
10319:
10315:
10304:
10301:
10298:
10290:
10286:
10282:
10277:
10273:
10261:
10257:
10253:
10248:
10244:
10233:
10230:
10227:
10219:
10215:
10211:
10206:
10202:
10190:
10186:
10182:
10177:
10173:
10162:
10159:
10154:
10150:
10146:
10141:
10137:
10133:
10128:
10124:
10120:
10115:
10111:
10103:
10095:
10094:
10093:
10087:
10078:
10069:
10060:
10054:
10048:
10042:
10036:
10032:
10025:
10018:
10011:
10004:
9994:
9987:
9980:
9973:
9968:
9959:
9952:
9945:
9938:
9927:
9920:
9913:
9906:
9895:
9888:
9881:
9874:
9868:
9867:
9866:
9855:
9848:
9841:
9834:
9827:
9820:
9813:
9806:
9799:
9788:
9781:
9774:
9770:
9766:
9759:
9752:
9748:
9741:
9732:
9723:
9714:
9708:
9704:
9700:
9696:
9692:the roots of
9691:
9690:
9689:
9682:
9678:
9674:
9665:
9656:
9652:
9648:
9644:
9640:
9631:
9622:
9618:
9613:
9609:
9605:
9601:
9597:
9593:
9589:
9585:
9581:
9577:
9573:
9572:
9571:
9568:
9564:
9560:
9556:
9551:
9547:
9537:
9535:
9529:
9523:
9519:
9514:
9513:
9512:
9504:
9471:
9467:
9463:
9460:
9455:
9451:
9447:
9444:
9441:
9438:
9435:
9428:
9424:
9420:
9417:
9412:
9408:
9404:
9401:
9398:
9395:
9392:
9385:
9382:
9379:
9376:
9369:
9361:
9360:
9359:
9356:
9350:
9344:
9342:
9333:
9326:
9324:
9310:
9305:
9301:
9297:
9294:
9288:
9285:
9282:
9277:
9273:
9266:
9261:
9257:
9253:
9250:
9247:
9242:
9238:
9230:
9229:
9226:
9225:
9220:
9216:
9192:
9187:
9183:
9179:
9176:
9174:
9166:
9163:
9158:
9154:
9150:
9147:
9145:
9134:
9131:
9122:
9119:
9111:
9107:
9103:
9101:
9084:
9081:
9078:
9072:
9069:
9066:
9063:
9051:
9048:
9045:
9039:
9036:
9033:
9030:
9019:
9015:
9011:
9009:
8999:
8991:
8988:
8985:
8977:
8973:
8969:
8964:
8956:
8953:
8950:
8942:
8938:
8934:
8932:
8925:
8921:
8917:
8912:
8902:
8898:
8894:
8891:
8883:
8879:
8867:
8866:
8865:
8833:
8830:
8827:
8824:
8814:
8811:
8808:
8802:
8799:
8796:
8789:
8786:
8783:
8780:
8775:
8771:
8767:
8764:
8757:
8749:
8748:
8747:
8744:
8740:
8730:
8724:
8717:
8713:
8707:
8703:
8699:
8695:
8691:
8665:
8662:
8659:
8656:
8649:
8646:
8643:
8640:
8637:
8634:
8631:
8624:
8621:
8618:
8615:
8612:
8609:
8606:
8603:
8596:
8593:
8590:
8587:
8584:
8577:
8569:
8568:
8567:
8565:
8542:
8539:
8536:
8533:
8527:
8524:
8521:
8518:
8515:
8509:
8504:
8500:
8493:
8490:
8487:
8484:
8481:
8478:
8472:
8467:
8463:
8456:
8453:
8450:
8444:
8439:
8435:
8431:
8429:
8418:
8415:
8412:
8409:
8406:
8401:
8397:
8387:
8384:
8381:
8378:
8375:
8370:
8366:
8359:
8357:
8352:
8349:
8346:
8343:
8340:
8335:
8331:
8327:
8324:
8319:
8315:
8311:
8308:
8303:
8299:
8287:
8286:
8285:
8277:
8274:
8268:
8259:
8240:
8235:
8228:
8220:
8214:
8209:
8199:
8195:
8191:
8188:
8185:
8182:
8179:
8175:
8171:
8164:
8160:
8154:
8151:
8144:
8140:
8133:
8125:
8121:
8117:
8111:
8107:
8101:
8098:
8095:
8088:
8087:
8086:
8083:
8037:
8032:
8025:
8017:
8011:
8006:
7996:
7992:
7988:
7985:
7982:
7979:
7976:
7972:
7968:
7961:
7957:
7951:
7948:
7941:
7937:
7930:
7927:
7920:
7919:
7918:
7916:
7897:
7894:
7890:
7881:
7878:
7873:
7869:
7864:
7861:
7856:
7853:
7848:
7845:
7842:
7837:
7834:
7829:
7824:
7820:
7815:
7810:
7801:
7798:
7793:
7789:
7784:
7781:
7776:
7773:
7768:
7765:
7762:
7757:
7754:
7749:
7744:
7740:
7735:
7727:
7726:
7725:
7723:
7722:
7721:
7713:
7709:
7705:
7701:
7693:
7689:
7683:
7679:
7659:
7654:
7649:
7640:
7637:
7632:
7628:
7623:
7618:
7615:
7610:
7606:
7601:
7596:
7591:
7587:
7584:
7579:
7576:
7571:
7566:
7562:
7557:
7548:
7547:
7546:
7544:
7543:
7542:
7534:
7524:
7520:
7513:
7506:
7499:
7479:
7474:
7469:
7460:
7457:
7452:
7448:
7443:
7440:
7435:
7432:
7426:
7417:
7416:
7415:
7413:
7412:
7411:
7401:
7393:
7384:
7377:
7375:
7361:
7358:
7353:
7349:
7345:
7342:
7336:
7333:
7330:
7325:
7321:
7317:
7311:
7306:
7302:
7298:
7295:
7292:
7287:
7283:
7279:
7272:
7271:
7268:
7250:
7247:
7244:
7240:
7236:
7233:
7228:
7223:
7219:
7213:
7210:
7207:
7204:
7199:
7195:
7190:
7183:
7180:
7174:
7171:
7166:
7158:
7155:
7145:
7144:
7143:
7137:
7129:
7125:
7116:
7105:
7098:
7096:
7082:
7079:
7076:
7071:
7066:
7062:
7056:
7053:
7050:
7047:
7042:
7038:
7034:
7031:
7028:
7025:
7020:
7016:
7012:
7009:
7006:
7001:
6996:
6992:
6989:
6984:
6981:
6976:
6971:
6967:
6962:
6953:
6952:
6949:
6942:
6938:
6934:
6931:
6906:
6901:
6896:
6892:
6886:
6883:
6880:
6877:
6874:
6871:
6868:
6863:
6858:
6852:
6849:
6844:
6839:
6835:
6830:
6821:
6820:
6819:
6817:
6798:
6795:
6792:
6789:
6786:
6783:
6780:
6775:
6771:
6767:
6764:
6759:
6755:
6747:
6746:
6745:
6744:
6734:
6723:
6711:
6700:
6684:
6673:
6664:
6640:
6632:
6625:
6621:
6615:
6608:
6604:
6598:
6591:
6587:
6579:
6575:
6571:
6568:
6563:
6556:
6552:
6544:
6540:
6534:
6530:
6526:
6523:
6518:
6511:
6507:
6499:
6495:
6491:
6488:
6483:
6476:
6472:
6466:
6463:
6457:
6452:
6448:
6443:
6439:
6435:
6432:
6429:
6426:
6423:
6420:
6417:
6414:
6411:
6406:
6402:
6398:
6395:
6389:
6387:
6382:
6370:
6363:
6359:
6353:
6346:
6339:
6335:
6327:
6323:
6319:
6316:
6311:
6307:
6301:
6297:
6291:
6287:
6283:
6280:
6275:
6268:
6264:
6255:
6250:
6246:
6243:
6240:
6237:
6234:
6231:
6228:
6223:
6219:
6212:
6210:
6205:
6193:
6186:
6182:
6176:
6169:
6162:
6158:
6152:
6149:
6144:
6140:
6134:
6130:
6126:
6120:
6115:
6109:
6105:
6101:
6098:
6095:
6092:
6086:
6084:
6079:
6068:
6067:
6066:
6062:
6058:
6054:
6050:
6030:
6022:
6009:
5998:
5990:
5977:
5966:
5958:
5945:
5934:
5926:
5913:
5902:
5894:
5887:
5883:
5879:
5875:
5871:
5862:
5856:
5839:
5836:
5831:
5827:
5823:
5820:
5815:
5811:
5807:
5802:
5798:
5792:
5788:
5784:
5779:
5775:
5769:
5765:
5761:
5756:
5752:
5746:
5742:
5734:
5733:
5732:
5729:
5726:
5711:
5709:
5703:
5699:
5692:
5688:
5684:
5674:
5667:
5663:
5657:
5653:
5647:
5643:
5640:produces the
5633:
5624:
5620:
5606:
5598:
5590:
5586:
5579:
5568:
5560:
5552:
5545:
5537:
5529:
5525:
5520:
5499:
5495:
5489:
5485:
5481:
5478:
5475:
5470:
5466:
5462:
5457:
5453:
5447:
5443:
5439:
5434:
5430:
5424:
5420:
5416:
5411:
5407:
5401:
5397:
5393:
5387:
5381:
5374:
5373:
5372:
5346:
5339:
5335:
5329:
5326:
5324:
5317:
5313:
5305:
5298:
5294:
5288:
5285:
5283:
5276:
5272:
5264:
5257:
5253:
5247:
5244:
5242:
5235:
5231:
5223:
5216:
5212:
5206:
5203:
5201:
5194:
5190:
5178:
5177:
5176:
5172:
5168:
5161:
5157:
5148:
5139:
5130:
5126:
5120:
5116:
5110:
5106:
5102:
5096:
5090:
5086:
5080:
5076:
5070:
5066:
5060:
5058:
5054:
5033:
5029:
5025:
5020:
5016:
5010:
5006:
5002:
4997:
4993:
4987:
4983:
4979:
4973:
4967:
4960:
4959:
4958:
4951:
4944:
4933:
4931:
4927:
4923:
4913:
4909:
4903:
4898:
4895:
4890:
4885:
4884:
4883:
4881:
4878:
4874:
4866:
4864:
4860:
4854:
4850:
4830:
4822:
4818:
4814:
4811:
4806:
4802:
4798:
4793:
4789:
4783:
4779:
4767:
4763:
4759:
4756:
4751:
4747:
4743:
4738:
4734:
4728:
4724:
4717:
4711:
4705:
4698:
4697:
4696:
4674:
4670:
4666:
4663:
4658:
4654:
4650:
4645:
4641:
4635:
4631:
4627:
4622:
4618:
4612:
4608:
4596:
4592:
4588:
4585:
4579:
4573:
4567:
4560:
4559:
4558:
4554:
4550:
4545:
4541:
4535:
4531:
4524:
4520:
4513:
4509:
4505:
4501:
4497:
4493:
4488:
4469:
4464:
4460:
4456:
4453:
4448:
4444:
4440:
4435:
4431:
4425:
4421:
4417:
4412:
4408:
4402:
4398:
4394:
4389:
4385:
4379:
4375:
4371:
4365:
4359:
4352:
4351:
4350:
4340:Simpler cases
4334:
4318:
4315:
4312:
4307:
4282:
4279:
4268:
4254:
4251:
4247:
4243:
4240:
4237:
4232:
4228:
4224:
4221:
4216:
4212:
4189:
4185:
4176:
4160:
4154:
4151:
4148:
4145:
4142:
4139:
4134:
4130:
4126:
4123:
4117:
4095:
4091:
4082:
4066:
4063:
4060:
4055:
4030:
4027:
4024:
4019:
3994:
3991:
3980:
3977:
3973:
3957:
3949:
3933:
3928:
3923:
3915:
3912:
3908:
3902:
3899:
3895:
3873:
3870:
3867:
3847:
3827:
3824:
3821:
3818:
3810:
3796:
3791:
3766:
3761:
3732:
3727:
3700:
3697:
3694:
3691:
3667:
3662:
3652:
3645:
3640:
3632:
3629:
3624:
3619:
3592:
3589:
3586:
3581:
3556:
3553:
3542:
3541:
3524:
3520:
3510:
3505:
3495:
3489:
3479:
3475:
3472:
3469:
3466:
3459:
3458:
3456:
3436:
3433:
3428:
3425:
3418:
3405:
3402:
3398:
3393:
3390:
3382:
3379:
3374:
3367:
3364:
3359:
3356:
3349:
3348:
3347:
3346:
3343:, as follows:
3342:
3338:
3322:
3319:
3299:
3279:
3272:the value of
3259:
3256:
3253:
3242:
3241:
3232:
3228:
3224:
3188:
3179:
3176:
3173:
3168:
3163:
3155:
3152:
3147:
3142:
3130:
3129:
3128:
3107:
3104:
3101:
3098:
3095:
3090:
3086:
3082:
3079:
3076:
3073:
3068:
3064:
3060:
3057:
3054:
3051:
3048:
3045:
3042:
3037:
3033:
3029:
3026:
3024:
3017:
3005:
3002:
2999:
2996:
2993:
2990:
2987:
2984:
2979:
2975:
2971:
2969:
2962:
2946:
2945:
2944:
2941:
2939:
2933:
2926:
2899:
2894:
2886:
2881:
2873:
2870:
2865:
2860:
2850:
2845:
2833:
2831:
2826:
2816:
2810:
2805:
2795:
2792:
2788:
2781:
2778:
2774:
2769:
2766:
2758:
2755:
2750:
2743:
2740:
2735:
2733:
2728:
2717:
2716:
2715:
2710:
2687:
2683:
2679:
2674:
2669:
2665:
2661:
2658:
2655:
2652:
2649:
2646:
2643:
2638:
2634:
2627:
2625:
2620:
2608:
2604:
2600:
2593:
2589:
2585:
2582:
2579:
2576:
2573:
2567:
2565:
2560:
2549:
2548:
2547:
2546:
2511:
2508:
2503:
2500:
2497:
2494:
2489:
2485:
2481:
2478:
2471:
2468:
2463:
2460:
2457:
2451:
2448:
2444:
2439:
2436:
2434:
2424:
2421:
2418:
2414:
2402:
2399:
2394:
2391:
2388:
2385:
2380:
2376:
2372:
2369:
2362:
2359:
2354:
2351:
2348:
2342:
2339:
2335:
2330:
2327:
2325:
2315:
2312:
2309:
2305:
2293:
2292:
2291:
2289:
2285:
2281:
2277:
2253:
2250:
2247:
2244:
2241:
2238:
2235:
2230:
2226:
2222:
2219:
2214:
2210:
2206:
2203:
2198:
2194:
2190:
2183:
2182:
2181:
2175:
2166:
2157:
2148:
2127:
2124:
2121:
2118:
2115:
2112:
2109:
2104:
2100:
2096:
2093:
2088:
2084:
2080:
2077:
2072:
2068:
2058:
2049:
2035:
2030:
2026:
2022:
2019:
2016:
2013:
2008:
1998:
1994:
1990:
1938:
1929:
1911:
1900:
1893:
1892:
1886:
1871:
1852:
1833:
1832:
1830:
1822:
1809:
1798:
1797:
1791:
1788:
1780:
1779:
1778:
1775:
1756:
1752:
1748:
1745:
1742:
1739:
1734:
1730:
1726:
1723:
1720:
1715:
1711:
1707:
1704:
1701:
1696:
1692:
1686:
1682:
1678:
1675:
1672:
1667:
1663:
1659:
1656:
1653:
1646:
1645:
1644:
1627:
1624:
1621:
1618:
1615:
1612:
1609:
1606:
1603:
1598:
1594:
1590:
1585:
1573:
1572:
1571:
1564:
1555:
1531:
1528:
1525:
1522:
1519:
1516:
1511:
1507:
1503:
1500:
1497:
1492:
1488:
1484:
1481:
1474:
1473:
1472:
1470:
1461:
1452:
1426:
1422:
1418:
1415:
1412:
1409:
1406:
1403:
1400:
1393:
1392:
1391:
1368:
1364:
1358:
1354:
1348:
1344:
1340:
1337:
1332:
1328:
1322:
1318:
1314:
1311:
1306:
1302:
1296:
1292:
1288:
1285:
1282:
1279:
1276:
1271:
1267:
1263:
1260:
1255:
1251:
1245:
1241:
1237:
1234:
1229:
1225:
1219:
1215:
1211:
1208:
1205:
1203:
1195:
1190:
1186:
1182:
1179:
1176:
1171:
1167:
1163:
1160:
1157:
1154:
1151:
1148:
1145:
1140:
1136:
1132:
1129:
1126:
1123:
1120:
1115:
1111:
1105:
1101:
1097:
1094:
1091:
1086:
1082:
1078:
1073:
1069:
1065:
1062:
1059:
1057:
1047:
1043:
1037:
1033:
1029:
1026:
1023:
1018:
1014:
1010:
1005:
1001:
997:
994:
989:
985:
979:
975:
969:
965:
961:
958:
953:
949:
945:
942:
937:
933:
929:
926:
921:
917:
911:
907:
903:
896:
882:
881:
880:
879:
873:
853:
850:
847:
844:
841:
838:
835:
830:
826:
822:
819:
814:
810:
806:
803:
798:
794:
790:
783:
782:
781:
768:
750:
735:
732:
727:
721:
716:
713:
707:
701:
698:
693:
690:
680:
679:
678:
676:
653:
645:
627:
625:
621:
620:Intersections
617:
615:
611:
607:
602:
600:
596:
592:
587:
585:
580:
578:
574:
569:
567:
562:
559:
554:
550:
545:
543:
539:
535:
531:
527:
523:
519:
515:
505:
503:
499:
495:
490:
488:
487:
482:
478:
474:
465:
463:
459:
455:
450:
448:
444:
440:
436:
417:
414:
411:
406:
402:
398:
395:
390:
386:
382:
379:
373:
367:
360:
359:
358:
356:
352:
348:
344:
339:
337:
333:
327:
307:
304:
301:
298:
295:
292:
289:
286:
281:
277:
273:
270:
265:
261:
257:
254:
249:
245:
241:
234:
233:
232:
230:
229:
223:
221:
217:
213:
209:
190:
187:
184:
181:
178:
175:
170:
166:
162:
159:
154:
150:
146:
143:
138:
134:
130:
127:
121:
115:
108:
107:
106:
104:
100:
96:
88:
84:
80:
76:
72:
68:
64:
60:
56:
53:
49:
44:
40:
36:
29:
22:
14353:Discriminant
14272:Multivariate
14223:
14080:
14074:
14052:(1): 28–30.
14049:
14043:
14004:(1): 51–57,
14001:
13995:
13989:
13957:
13948:
13916:
13907:
13885:(1): 67–70.
13882:
13876:
13863:
13831:
13822:
13803:
13799:
13789:
13767:(2): 51–55.
13764:
13760:
13754:
13730:
13724:
13718:
13698:
13691:
13667:
13661:
13652:
13626:
13622:
13616:
13602:
13588:
13576:. Retrieved
13572:
13562:
13553:
13541:
13533:
13528:
13508:
13497:
13483:
13469:
13457:. Retrieved
13453:
13443:
13419:
13412:
13367:
13361:
13348:
13341:
13338:
13329:
13322:
13315:
13306:
13299:
13292:
13283:
13276:
13269:
13221:
13215:
13209:
13200:
13193:
13190:
12996:
12992:
12989:
12980:
12970:
12963:
12956:
12946:
12942:
12935:
12931:
12924:
12920:
12916:
12912:
12901:
12897:
12893:
12889:
12886:
12879:
12867:
12857:
12854:
12743:
12511:
12510:
12505:
12498:
12489:
12480:
12471:
12461:
12458:
12447:
12336:
12323:
12313:
12304:
12295:
12292:
12146:
12137:
12128:
12112:
12101:
12092:
12085:
11695:. If we set
11691:
11687:
11683:
11679:
11675:
11661:
11652:
11635:
11619:
11613:
11597:
11595:) only when
11589:
11588:
11579:is equal to
11575:
11569:
11566:
11514:
11507:
11500:
11492:
11487:square root
11484:
11479:
11471:
11466:square root
11463:
11457:
11448:
11438:
11428:
11418:
11411:
11407:
11397:
11391:
11385:
11378:
11372:
11366:
11359:
11353:
11347:
11340:
11334:
11328:
11322:
11314:
11306:
11294:
11288:
11280:
11272:
11263:
11255:
11254:
11249:
11243:
11237:
11229:
11221:
11213:
11201:
11192:
11183:
11174:
11168:
11162:
11156:
11153:
11144:
11137:
11130:
11123:
11113:
11106:
11099:
11092:
11078:
11071:
11064:
11057:
11049:
10792:
10577:
10568:
10559:
10550:
10547:
10540:
10531:
10524:
10516:
10507:
10500:
10492:
10483:
10476:
10468:
10459:
10452:
10443:
10434:
10427:
10421:
10412:
10405:
10402:
10085:
10076:
10067:
10058:
10052:
10046:
10040:
10034:
10030:
10023:
10016:
10009:
10002:
9992:
9985:
9978:
9971:
9957:
9950:
9943:
9936:
9925:
9918:
9911:
9904:
9893:
9886:
9879:
9872:
9863:
9862:
9853:
9846:
9839:
9832:
9825:
9818:
9811:
9804:
9797:
9793:
9786:
9779:
9772:
9764:
9757:
9750:
9739:
9730:
9721:
9712:
9706:
9702:
9698:
9694:
9687:
9680:
9676:
9672:
9663:
9654:
9646:
9642:
9638:
9629:
9620:
9611:
9607:
9603:
9599:
9595:
9591:
9587:
9583:
9579:
9575:
9566:
9562:
9558:
9554:
9549:
9543:
9527:
9521:
9517:
9509:
9508:
9505:
9493:
9354:
9348:
9345:
9338:
9327:
9218:
9214:
9211:
8855:
8742:
8738:
8728:
8722:
8715:
8711:
8705:
8701:
8697:
8693:
8687:
8561:
8283:
8272:
8266:
8255:
8084:
8052:
7912:
7718:
7717:
7711:
7707:
7703:
7699:
7691:
7687:
7681:
7677:
7674:
7539:
7538:
7537:, equation (
7532:
7525:
7518:
7511:
7504:
7497:
7494:
7408:
7407:
7390:This is the
7389:
7378:
7266:
7128:discriminant
7117:
7110:
7099:
6940:
6936:
6932:
6929:
6921:
6813:
6742:
6740:
6721:
6709:
6698:
6682:
6671:
6662:
6659:
6060:
6056:
6052:
6048:
6028:
6020:
6007:
5996:
5988:
5975:
5964:
5956:
5943:
5932:
5924:
5911:
5900:
5892:
5885:
5881:
5877:
5873:
5869:
5860:
5858:Dividing by
5857:
5854:
5730:
5724:
5722:
5701:
5697:
5690:
5686:
5682:
5672:
5665:
5661:
5655:
5651:
5645:
5631:
5622:
5618:
5604:
5596:
5588:
5584:
5577:
5566:
5558:
5550:
5543:
5535:
5527:
5523:
5516:
5370:
5170:
5166:
5159:
5155:
5146:
5137:
5128:
5124:
5118:
5114:
5108:
5104:
5100:
5094:
5088:
5078:
5074:
5068:
5064:
5061:
5056:
5052:
5051:is called a
5050:
4949:
4942:
4939:
4919:
4911:
4907:
4901:
4888:
4876:
4872:
4867:
4852:
4848:
4845:
4694:
4552:
4548:
4533:
4529:
4522:
4518:
4511:
4507:
4503:
4499:
4495:
4491:
4484:
4348:
3605:the sign of
3226:
3223:discriminant
3126:
2942:
2931:
2924:
2921:
2713:
2534:
2269:
2173:
2164:
2155:
2146:
2143:
2060:Solution of
1950:
1936:
1927:
1861:> 0 and (
1776:
1773:
1642:
1562:
1553:
1546:
1459:
1450:
1443:
1389:
878:discriminant
871:
868:
779:
765:
633:
618:
603:
588:
582:Finding the
581:
576:
570:
563:
557:
546:
511:
508:Applications
491:
484:
483:in the book
471:
453:
451:
446:
438:
432:
353:refers to a
350:
346:
342:
340:
325:
322:
226:
224:
219:
207:
205:
105:of the form
98:
92:
86:
82:
78:
70:
58:
39:
14302:Homogeneous
14297:Square-free
14292:Irreducible
14157:Polynomials
14083:: 271–275.
9794:Therefore,
5519:palindromic
1905:> 0 and
1891:= 0, then:
1857:> 0 or (
1842:< 0 and
1838:< 0 and
1803:< 0 and
652:secant line
591:eigenvalues
571:In optics,
343:biquadratic
14373:Categories
14262:Univariate
14115:PlanetMath
13982:0557.01014
13941:0724.12001
13856:51.0020.07
13435:References
12088:involution
11665:from
9212:If we set
7545:) becomes
5517:is almost
5083:becomes a
2714:and where
1979:= 0 then
1814:> 0 or
1547:such that
1444:such that
514:coordinate
332:derivative
212:polynomial
14384:Equations
14348:Resultant
14287:Trinomial
14267:Bivariate
14097:124512391
13513:, Dover,
13505:(1993) ,
13162:−
12774:×
12668:−
12610:−
12421:−
12399:−
12330:. By the
12253:−
12215:−
12177:−
12038:−
12025:−
11960:−
11947:−
11869:−
11843:−
11546:β
11539:α
11528:−
11015:γ
11005:β
11000:−
10995:α
10990:−
10959:γ
10954:−
10949:β
10939:α
10934:−
10903:γ
10898:−
10893:β
10888:−
10883:α
10850:γ
10840:β
10830:α
10764:γ
10727:β
10690:α
10425:and that
10376:γ
10373:−
10305:β
10302:−
10234:α
10231:−
9461:−
9383:−
9339:which is
9298:−
9283:−
9082:−
9073:−
9049:−
8989:−
8970:−
8918:−
8812:−
8736:, we get
8196:±
8172:−
8161:±
8141:±
8102:−
7993:±
7969:−
7958:±
7938:±
7849:−
7785:−
7624:−
7444:−
7346:−
7331:−
7234:−
7172:−
7156:−
7077:−
7026:−
6881:−
6872:−
6696:(that is
6524:−
6464:−
6421:−
6396:−
6281:−
6229:−
6150:−
6099:−
5340:−
5330:−
5299:−
5248:−
5085:quadratic
4589:−
4487:reducible
4333:reducible
4313:≠
4304:Δ
4277:Δ
4241:−
4052:Δ
4016:Δ
3989:Δ
3948:numerator
3871:≠
3788:Δ
3758:Δ
3724:Δ
3695:≠
3659:Δ
3637:Δ
3630:−
3616:Δ
3578:Δ
3554:≠
3551:Δ
3502:Δ
3486:Δ
3476:
3467:φ
3434:φ
3429:
3415:Δ
3375:−
3251:Δ
3209:Δ
3183:Δ
3177:−
3160:Δ
3153:−
3139:Δ
3096:−
3043:−
3014:Δ
2985:−
2959:Δ
2940:, below)
2878:Δ
2871:−
2857:Δ
2842:Δ
2802:Δ
2751:−
2644:−
2583:−
2504:−
2495:−
2479:−
2464:±
2440:−
2386:−
2370:−
2355:±
2349:−
2331:−
2284:quadratic
2005:Δ
1746:−
1724:−
1676:−
1604:−
1582:Δ
1517:−
1416:−
1312:−
1286:−
1235:−
1206:−
1124:−
1092:−
1027:−
959:−
927:−
894:Δ
736:φ
673:into the
593:of a 4×4
486:Ars Magna
50:and four
14333:Division
14282:Binomial
14277:Monomial
13899:53375377
13424:variable
13375:See also
11669:to
11235:. Since
8732:is
7526:Now, if
7134:of this
6065:, where
5680:. Since
4540:rational
4516:, where
3335:this is
1963:= 0 and
1829:multiple
1794:∆ > 0
1783:∆ < 0
771:Solution
669:divides
661:than to
634:Letting
624:quadrics
458:radicals
435:infinity
103:function
14066:2688990
14026:1369151
14018:2975214
13918:Algebra
13781:2972804
13747:2305030
13684:2589403
13645:1847825
13578:27 July
13459:27 July
13313:, and
12321:in the
12123:
12109:
11512:√
11490:√
11469:√
11320:√
11312:√
11304:√
11286:√
11278:√
11270:√
11227:√
11219:√
11211:√
11052:8 (= 2)
8746:, and:
8263:√
7402:. When
6729:
6706:
6693:
6679:
6039:
6025:
6015:
5993:
5983:
5961:
5951:
5929:
5919:
5897:
5890:, with
5710:twice.
5636:
5615:
5609:
5593:
5571:
5555:
5548:
5532:
5072:. Then
1941:
1924:
1865:≠ 0 or
1567:
1550:
1464:
1447:
665:, then
553:endmill
468:History
454:quartic
347:quartic
95:algebra
75:maximum
63:complex
21:Quantic
14170:degree
14095:
14064:
14024:
14016:
13980:
13970:
13939:
13929:
13897:
13854:
13844:
13779:
13745:
13706:
13682:
13643:
13517:
11659:, for
11583:, but
11508:define
11445:, and
11247:, and
11199:, and
11166:, and
10575:, and
10083:, and
10050:, and
9933:, and
9737:, and
8053:where
5986:, and
5135:. Let
4485:It is
3473:arccos
3457:where
3388:
3201:where
3186:
2936:, see
2764:
2535:where
2430:
2321:
2171:, and
1973:> 0
1957:> 0
595:matrix
542:optics
330:. The
323:where
216:degree
206:where
67:minima
14093:S2CID
14062:JSTOR
14014:JSTOR
13895:S2CID
13873:(PDF)
13838:Dover
13777:JSTOR
13743:JSTOR
13680:JSTOR
13550:(PDF)
13406:Notes
13355:as a
12744:then
11630:as a
11462:pick
9824:) = −
9688:then
9546:Euler
7714:) = 0
5704:) = 0
5521:, as
4544:field
3976:below
2943:with
2270:with
1825:∆ = 0
1469:below
522:torus
512:Each
477:cubic
437:. If
101:is a
55:roots
14159:and
13968:ISBN
13927:ISBN
13842:ISBN
13704:ISBN
13580:2020
13515:ISBN
13461:2020
13365:and
13213:and
12929:and
12144:and
11614:The
11483:and
9710:are
9661:and
9627:and
9550:some
8860:and
8720:for
8060:and
6042:for
5731:Let
5530:) =
5175:are
5144:and
5107:) =
4527:and
4498:) =
4295:and
4007:and
3569:and
3254:>
3127:and
2922:(if
2539:and
2286:and
1975:and
1880:and
638:and
589:The
575:is "
540:and
97:, a
52:real
14168:By
14113:at
14085:doi
14054:doi
14006:doi
14002:103
13978:Zbl
13937:Zbl
13887:doi
13852:JFM
13808:doi
13769:doi
13735:doi
13672:doi
13668:105
13631:doi
13357:3×3
12962:≔ (
12938:= 0
12927:= 0
12904:= 0
12339:= 0
12107:= −
11600:= 0
11573:or
11520:as
11498:of
11485:any
11477:of
11464:any
11293:= ±
11151:}.
11143:, −
11136:, −
11129:, −
10029:= −
9998:= 0
9763:= −
9590:= (
9530:= 0
9520:− 4
9346:If
8741:= −
8562:By
8271:= 2
8071:or
7696:or
7694:= 0
7535:≠ 0
7521:= 0
7514:≠ 0
7507:= 0
7500:= 0
7130:in
6660:If
6616:256
6492:256
6453:256
6415:256
6063:= 0
5888:= 0
5693:= 0
5678:= 0
5671:− 2
5638:= 0
5612:in
5580:= 1
5092:in
4955:= 0
4940:If
4861:or
4695:or
4489:if
4269:If
3981:If
3950:of
3811:If
3749:as
3543:If
3426:cos
3243:If
2934:= 0
2929:or
2927:= 0
1918:= 0
1912:If
1901:If
1894:If
1887:If
1878:= 0
1872:If
1853:If
1848:≠ 0
1834:If
1823:If
1810:If
1799:If
1792:If
1781:If
1471:);
1063:144
998:144
962:128
930:192
904:256
874:≠ 0
608:or
547:In
328:≠ 0
214:of
93:In
14375::
14091:.
14081:96
14079:.
14060:.
14050:39
14048:.
14022:MR
14020:,
14012:,
14000:,
13976:,
13966:,
13962:,
13935:,
13925:,
13893:.
13883:80
13881:.
13875:.
13850:,
13840:,
13836:,
13802:.
13798:.
13775:.
13765:29
13763:.
13741:,
13731:56
13729:,
13678:,
13666:,
13641:MR
13639:,
13627:15
13625:,
13607:,
13601:,
13597:,
13571:.
13552:.
13488:,
13482:,
13478:,
13452:.
13414:^α
13349:μF
13347:+
13342:λF
13333:23
13328:+
13326:14
13321:=
13310:24
13305:+
13303:13
13298:=
13290:,
13287:34
13282:+
13280:12
13275:=
13201:μF
13199:+
13194:λF
13153::=
13058::=
12995:=
12969:,
12945:=
12934:−
12923:+
12921:qx
12919:+
12917:py
12915:+
12900:+
12898:qx
12896:+
12894:px
12892:+
12872:.
12865:.
12496:.
12135:,
11690:+
11688:dx
11686:+
11684:cx
11682:+
11680:bx
11678:+
11435:,
11425:,
11384:+
11365:+
11346:+
11327:=
11241:,
11190:,
11181:,
11160:,
11122:{−
11112:,
11105:,
11098:,
11077:,
11070:,
11063:,
10566:,
10557:,
10530:+
10506:+
10482:+
10458:+
10433:+
10411:+
10074:,
10065:,
10044:,
10033:+
10022:+
10015:+
10008:+
9991:+
9984:+
9977:+
9956:+
9949:)(
9942:+
9935:−(
9924:+
9917:)(
9910:+
9903:−(
9901:,
9892:+
9885:)(
9878:+
9871:−(
9852:+
9845:)(
9838:+
9831:−(
9817:+
9810:)(
9803:+
9785:=
9778:+
9756:+
9728:,
9719:,
9705:+
9703:qx
9701:+
9699:px
9697:+
9679:+
9677:sx
9675:−
9645:+
9643:sx
9641:+
9610:+
9608:sx
9606:−
9602:)(
9598:+
9596:sx
9594:+
9586:+
9584:qx
9582:+
9580:px
9578:+
9565:+
9563:qx
9561:+
9559:px
9557:+
9536:.
9217:=
8718:/4
8714:−
8704:+
8702:qy
8700:+
8698:py
8696:+
8276:.
7898:0.
7710:−
7706:)(
7702:+
7690:−
7680:=
7523:.
7381:1a
7362:0.
7115:.
6939:+
6937:pm
6935:+
6799:0.
6704:−
6677:−
6572:16
6527:64
6436:16
6424:64
6059:+
6057:qy
6055:+
6053:py
6051:+
6023:−
5991:=
5959:=
5954:,
5927:=
5922:,
5895:=
5884:+
5882:dx
5880:+
5878:cx
5876:+
5874:bx
5872:+
5689:+
5687:xz
5685:−
5673:ma
5664:+
5654:+
5591:+
5587:=
5528:mx
5127:+
5117:+
5098::
5067:=
4948:=
4924:,
4865:.
4506:)×
3874:0.
3180:27
3099:72
3080:27
3061:27
3000:12
2290:.
2162:,
2153:,
1991:16
1959:,
1727:16
1705:16
1679:16
1660:64
1619:12
1264:18
1238:27
1180:16
1155:18
1127:80
1030:27
677::
671:FH
655:FG
536:,
532:,
489:.
464:.
338:.
225:A
222:.
14149:e
14142:t
14135:v
14117:.
14099:.
14087::
14068:.
14056::
14008::
13901:.
13889::
13816:.
13810::
13804:5
13783:.
13771::
13737::
13712:.
13674::
13633::
13582:.
13556:.
13463:.
13420:e
13368:μ
13362:λ
13352:2
13345:1
13330:L
13323:L
13319:3
13316:Q
13307:L
13300:L
13296:2
13293:Q
13284:L
13277:L
13273:1
13270:Q
13265:6
13247:)
13242:2
13239:4
13234:(
13216:μ
13210:λ
13204:2
13197:1
13170:2
13166:X
13159:Z
13156:Y
13146:)
13143:Z
13140:,
13137:Y
13134:,
13131:X
13128:(
13123:2
13119:F
13111:,
13106:2
13102:Z
13098:r
13095:+
13092:Z
13089:X
13086:q
13083:+
13080:Z
13077:Y
13074:p
13071:+
13066:2
13062:Y
13051:)
13048:Z
13045:,
13042:Y
13039:,
13036:X
13033:(
13028:1
13024:F
12997:x
12993:y
12983:i
12981:x
12976:)
12973:i
12971:x
12966:i
12964:x
12959:i
12957:P
12947:x
12943:y
12936:x
12932:y
12925:r
12913:y
12908:x
12902:r
12890:x
12858:s
12840:.
12837:e
12834:+
12831:x
12828:d
12825:+
12820:2
12816:x
12812:c
12809:+
12804:4
12800:x
12796:=
12793:)
12790:x
12787:(
12782:2
12778:F
12771:)
12768:x
12765:(
12760:1
12756:F
12722:s
12719:2
12715:d
12710:+
12705:2
12700:2
12696:s
12690:+
12685:2
12682:c
12677:+
12674:x
12671:s
12663:2
12659:x
12655:=
12648:)
12645:x
12642:(
12637:2
12633:F
12622:s
12619:2
12615:d
12605:2
12600:2
12596:s
12590:+
12585:2
12582:c
12577:+
12574:x
12571:s
12568:+
12563:2
12559:x
12555:=
12548:)
12545:x
12542:(
12537:1
12533:F
12513:3
12506:s
12492:i
12490:x
12483:i
12481:s
12474:i
12472:x
12462:s
12452:)
12450:3
12448:(
12429:2
12425:d
12416:2
12412:s
12408:)
12405:e
12402:4
12394:2
12390:c
12386:(
12383:+
12378:4
12374:s
12370:c
12367:2
12364:+
12359:6
12355:s
12337:b
12326:i
12324:x
12314:s
12307:i
12305:x
12298:i
12296:s
12278:.
12275:)
12270:2
12263:3
12259:s
12248:2
12244:s
12240:(
12237:)
12232:2
12225:2
12221:s
12210:2
12206:s
12202:(
12199:)
12194:2
12187:1
12183:s
12172:2
12168:s
12164:(
12150:3
12147:s
12141:2
12138:s
12132:1
12129:s
12120:2
12117:/
12113:b
12105:0
12102:s
12095:i
12093:s
12067:,
12064:)
12059:3
12055:x
12051:+
12046:2
12042:x
12033:1
12029:x
12020:0
12016:x
12012:(
12006:2
12003:1
11997:=
11988:3
11984:s
11976:,
11973:)
11968:3
11964:x
11955:2
11951:x
11942:1
11938:x
11934:+
11929:0
11925:x
11921:(
11915:2
11912:1
11906:=
11897:2
11893:s
11885:,
11882:)
11877:3
11873:x
11864:2
11860:x
11856:+
11851:1
11847:x
11838:0
11834:x
11830:(
11824:2
11821:1
11815:=
11806:1
11802:s
11794:,
11791:)
11786:3
11782:x
11778:+
11773:2
11769:x
11765:+
11760:1
11756:x
11752:+
11747:0
11743:x
11739:(
11733:2
11730:1
11724:=
11715:0
11711:s
11692:e
11676:x
11671:3
11667:0
11662:i
11655:i
11653:x
11623:4
11620:S
11598:q
11591:2
11585:0
11581:0
11576:β
11570:α
11563:.
11533:q
11515:γ
11504:;
11501:β
11493:β
11480:α
11472:α
11452:4
11449:r
11447:−
11442:3
11439:r
11437:−
11432:2
11429:r
11427:−
11422:1
11419:r
11417:−
11412:q
11410:−
11403:.
11401:4
11398:r
11395:3
11392:r
11389:2
11386:r
11382:4
11379:r
11376:3
11373:r
11370:1
11367:r
11363:4
11360:r
11357:2
11354:r
11351:1
11348:r
11344:3
11341:r
11338:2
11335:r
11332:1
11329:r
11323:γ
11315:β
11307:α
11295:q
11289:γ
11281:β
11273:α
11264:q
11257:2
11250:γ
11244:β
11238:α
11230:γ
11222:β
11214:α
11205:4
11202:r
11196:3
11193:r
11187:2
11184:r
11178:1
11175:r
11169:γ
11163:β
11157:α
11148:4
11145:r
11141:3
11138:r
11134:2
11131:r
11127:1
11124:r
11117:4
11114:r
11110:3
11107:r
11103:2
11100:r
11096:1
11093:r
11091:{
11087:2
11082:4
11079:r
11075:3
11072:r
11068:2
11065:r
11061:1
11058:r
11056:{
11027:.
11021:2
11010:+
10984:=
10979:4
10975:r
10965:2
10944:+
10928:=
10923:3
10919:r
10909:2
10875:=
10870:2
10866:r
10856:2
10845:+
10835:+
10822:=
10817:1
10813:r
10805:{
10770:;
10759:=
10754:4
10750:r
10746:+
10741:1
10737:r
10722:=
10717:3
10713:r
10709:+
10704:1
10700:r
10685:=
10680:2
10676:r
10672:+
10667:1
10663:r
10655:0
10652:=
10647:4
10643:r
10639:+
10634:3
10630:r
10626:+
10621:2
10617:r
10613:+
10608:1
10604:r
10596:{
10581:4
10578:r
10572:3
10569:r
10563:2
10560:r
10554:1
10551:r
10544:.
10541:γ
10535:3
10532:r
10528:2
10525:r
10520:,
10517:γ
10511:4
10508:r
10504:1
10501:r
10496:,
10493:β
10487:4
10484:r
10480:2
10477:r
10472:,
10469:β
10463:3
10460:r
10456:1
10453:r
10444:α
10438:4
10435:r
10431:3
10428:r
10422:α
10416:2
10413:r
10409:1
10406:r
10380:.
10370:=
10367:)
10362:3
10358:r
10354:+
10349:2
10345:r
10341:(
10338:)
10333:4
10329:r
10325:+
10320:1
10316:r
10312:(
10299:=
10296:)
10291:4
10287:r
10283:+
10278:2
10274:r
10270:(
10267:)
10262:3
10258:r
10254:+
10249:1
10245:r
10241:(
10228:=
10225:)
10220:4
10216:r
10212:+
10207:3
10203:r
10199:(
10196:)
10191:2
10187:r
10183:+
10178:1
10174:r
10170:(
10163:0
10160:=
10155:4
10151:r
10147:+
10142:3
10138:r
10134:+
10129:2
10125:r
10121:+
10116:1
10112:r
10104:{
10089:4
10086:r
10080:3
10077:r
10071:2
10068:r
10062:1
10059:r
10053:γ
10047:β
10041:α
10035:s
10031:s
10027:4
10024:r
10020:3
10017:r
10013:2
10010:r
10006:1
10003:r
9996:4
9993:r
9989:3
9986:r
9982:2
9979:r
9975:1
9972:r
9963:)
9961:3
9958:r
9954:2
9951:r
9947:4
9944:r
9940:1
9937:r
9931:)
9929:4
9926:r
9922:2
9919:r
9915:3
9912:r
9908:1
9905:r
9899:)
9897:4
9894:r
9890:3
9887:r
9883:2
9880:r
9876:1
9873:r
9865:2
9859:)
9857:4
9854:r
9850:3
9847:r
9843:2
9840:r
9836:1
9833:r
9826:s
9822:4
9819:r
9815:3
9812:r
9808:2
9805:r
9801:1
9798:r
9796:(
9790:.
9787:s
9783:4
9780:r
9776:3
9773:r
9768:,
9765:s
9761:2
9758:r
9754:1
9751:r
9746:,
9743:4
9740:r
9734:3
9731:r
9725:2
9722:r
9716:1
9713:r
9707:r
9695:x
9684:,
9681:v
9673:x
9667:4
9664:r
9658:3
9655:r
9650:,
9647:t
9639:x
9633:2
9630:r
9624:1
9621:r
9616:,
9614:)
9612:v
9604:x
9600:t
9592:x
9588:r
9576:x
9567:r
9555:x
9528:q
9522:r
9518:p
9511:2
9507:(
9501:U
9497:u
9472:u
9468:/
9464:q
9456:2
9452:u
9448:+
9445:p
9442:=
9439:v
9436:2
9429:u
9425:/
9421:q
9418:+
9413:2
9409:u
9405:+
9402:p
9399:=
9396:t
9393:2
9386:u
9380:=
9377:s
9370:{
9355:x
9349:u
9332:)
9330:2
9328:(
9311:,
9306:2
9302:q
9295:U
9292:)
9289:r
9286:4
9278:2
9274:p
9270:(
9267:+
9262:2
9258:U
9254:p
9251:2
9248:+
9243:3
9239:U
9219:u
9215:U
9193:r
9188:2
9184:u
9180:4
9177:=
9167:v
9164:t
9159:2
9155:u
9151:4
9148:=
9138:)
9135:v
9132:2
9129:(
9126:)
9123:t
9120:2
9117:(
9112:2
9108:u
9104:=
9094:]
9091:)
9088:)
9085:v
9079:t
9076:(
9070:v
9067:+
9064:t
9061:(
9058:)
9055:)
9052:v
9046:t
9043:(
9040:+
9037:v
9034:+
9031:t
9028:(
9025:[
9020:2
9016:u
9012:=
9000:2
8996:)
8992:v
8986:t
8983:(
8978:2
8974:u
8965:2
8961:)
8957:v
8954:+
8951:t
8948:(
8943:2
8939:u
8935:=
8926:2
8922:q
8913:2
8909:)
8903:2
8899:u
8895:+
8892:p
8889:(
8884:2
8880:u
8862:v
8858:t
8834:v
8831:t
8828:=
8825:r
8818:)
8815:v
8809:t
8806:(
8803:u
8800:=
8797:q
8790:v
8787:+
8784:t
8781:=
8776:2
8772:u
8768:+
8765:p
8758:{
8743:u
8739:s
8734:0
8729:y
8723:x
8716:b
8712:y
8706:r
8694:y
8666:v
8663:t
8660:=
8657:e
8650:u
8647:t
8644:+
8641:v
8638:s
8635:=
8632:d
8625:u
8622:s
8619:+
8616:v
8613:+
8610:t
8607:=
8604:c
8597:u
8594:+
8591:s
8588:=
8585:b
8578:{
8543:v
8540:t
8537:+
8534:x
8531:)
8528:u
8525:t
8522:+
8519:v
8516:s
8513:(
8510:+
8505:2
8501:x
8497:)
8494:u
8491:s
8488:+
8485:v
8482:+
8479:t
8476:(
8473:+
8468:3
8464:x
8460:)
8457:u
8454:+
8451:s
8448:(
8445:+
8440:4
8436:x
8432:=
8422:)
8419:v
8416:+
8413:x
8410:u
8407:+
8402:2
8398:x
8394:(
8391:)
8388:t
8385:+
8382:x
8379:s
8376:+
8371:2
8367:x
8363:(
8360:=
8353:e
8350:+
8347:x
8344:d
8341:+
8336:2
8332:x
8328:c
8325:+
8320:3
8316:x
8312:b
8309:+
8304:4
8300:x
8273:S
8267:m
8265:2
8241:.
8236:2
8229:)
8221:m
8215:q
8210:2
8200:1
8192:m
8189:2
8186:+
8183:p
8180:2
8176:(
8165:2
8155:m
8152:2
8145:1
8134:+
8126:4
8122:a
8118:4
8112:3
8108:a
8099:=
8096:x
8079:1
8077:±
8073:−
8069:+
8064:2
8062:±
8057:1
8055:±
8038:,
8033:2
8026:)
8018:m
8012:q
8007:2
7997:1
7989:m
7986:2
7983:+
7980:p
7977:2
7973:(
7962:2
7952:m
7949:2
7942:1
7931:=
7928:y
7895:=
7891:)
7882:m
7879:2
7874:2
7870:q
7865:+
7862:y
7857:m
7854:2
7846:m
7843:+
7838:2
7835:p
7830:+
7825:2
7821:y
7816:(
7811:)
7802:m
7799:2
7794:2
7790:q
7782:y
7777:m
7774:2
7769:+
7766:m
7763:+
7758:2
7755:p
7750:+
7745:2
7741:y
7736:(
7720:1
7712:N
7708:M
7704:N
7700:M
7698:(
7692:N
7688:M
7682:N
7678:M
7660:.
7655:2
7650:)
7641:m
7638:2
7633:2
7629:q
7619:m
7616:2
7611:y
7607:(
7602:=
7597:2
7592:)
7588:m
7585:+
7580:2
7577:p
7572:+
7567:2
7563:y
7558:(
7541:1
7533:m
7528:m
7519:y
7512:m
7505:q
7498:m
7480:.
7475:2
7470:)
7461:m
7458:2
7453:2
7449:q
7441:y
7436:m
7433:2
7427:(
7410:1
7404:m
7396:m
7383:)
7379:(
7359:=
7354:2
7350:q
7343:m
7340:)
7337:r
7334:8
7326:2
7322:p
7318:2
7315:(
7312:+
7307:2
7303:m
7299:p
7296:8
7293:+
7288:3
7284:m
7280:8
7251:,
7248:0
7245:=
7241:)
7237:r
7229:4
7224:2
7220:p
7214:+
7211:m
7208:p
7205:+
7200:2
7196:m
7191:(
7187:)
7184:m
7181:2
7178:(
7175:4
7167:2
7163:)
7159:q
7153:(
7140:m
7132:y
7120:m
7113:m
7104:)
7102:1
7100:(
7083:,
7080:r
7072:4
7067:2
7063:p
7057:+
7054:p
7051:m
7048:+
7043:2
7039:m
7035:+
7032:y
7029:q
7021:2
7017:y
7013:m
7010:2
7007:=
7002:2
6997:)
6993:m
6990:+
6985:2
6982:p
6977:+
6972:2
6968:y
6963:(
6946:y
6941:m
6933:m
6930:y
6928:2
6924:m
6907:.
6902:4
6897:2
6893:p
6887:+
6884:r
6878:y
6875:q
6869:=
6864:2
6859:)
6853:2
6850:p
6845:+
6840:2
6836:y
6831:(
6796:=
6793:r
6790:+
6787:y
6784:q
6781:+
6776:2
6772:y
6768:p
6765:+
6760:4
6756:y
6731:)
6725:4
6722:a
6720:4
6717:/
6713:3
6710:a
6702:0
6699:y
6690:4
6687:/
6683:b
6675:0
6672:y
6666:0
6663:y
6641:.
6633:4
6626:4
6622:a
6609:4
6605:a
6599:2
6592:3
6588:a
6580:2
6576:a
6569:+
6564:2
6557:4
6553:a
6545:3
6541:a
6535:1
6531:a
6519:3
6512:4
6508:a
6500:0
6496:a
6489:+
6484:4
6477:3
6473:a
6467:3
6458:=
6449:c
6444:2
6440:b
6433:+
6430:d
6427:b
6418:e
6412:+
6407:4
6403:b
6399:3
6390:=
6383:r
6371:3
6364:4
6360:a
6354:8
6347:2
6340:4
6336:a
6328:1
6324:a
6320:8
6317:+
6312:4
6308:a
6302:3
6298:a
6292:2
6288:a
6284:4
6276:3
6269:3
6265:a
6256:=
6251:8
6247:d
6244:8
6241:+
6238:c
6235:b
6232:4
6224:3
6220:b
6213:=
6206:q
6194:2
6187:4
6183:a
6177:8
6170:2
6163:3
6159:a
6153:3
6145:4
6141:a
6135:2
6131:a
6127:8
6121:=
6116:8
6110:2
6106:b
6102:3
6096:c
6093:8
6087:=
6080:p
6061:r
6049:y
6044:x
6036:4
6033:/
6029:b
6021:y
6011:4
6008:a
6004:/
6000:0
5997:a
5989:e
5979:4
5976:a
5972:/
5968:1
5965:a
5957:d
5947:4
5944:a
5940:/
5936:2
5933:a
5925:c
5915:4
5912:a
5908:/
5904:3
5901:a
5893:b
5886:e
5870:x
5864:4
5861:a
5840:0
5837:=
5832:0
5828:a
5824:+
5821:x
5816:1
5812:a
5808:+
5803:2
5799:x
5793:2
5789:a
5785:+
5780:3
5776:x
5770:3
5766:a
5762:+
5757:4
5753:x
5747:4
5743:a
5702:x
5700:(
5698:P
5691:m
5683:x
5676:0
5669:2
5666:a
5662:z
5659:1
5656:a
5652:z
5649:0
5646:a
5632:x
5628:/
5625:)
5623:x
5621:(
5619:P
5605:x
5601:/
5597:m
5589:x
5585:z
5578:m
5573:)
5567:x
5563:/
5559:m
5553:(
5551:P
5544:m
5540:/
5536:x
5526:(
5524:P
5500:2
5496:m
5490:0
5486:a
5482:+
5479:x
5476:m
5471:1
5467:a
5463:+
5458:2
5454:x
5448:2
5444:a
5440:+
5435:3
5431:x
5425:1
5421:a
5417:+
5412:4
5408:x
5402:0
5398:a
5394:=
5391:)
5388:x
5385:(
5382:P
5347:.
5336:z
5327:=
5318:4
5314:x
5306:,
5295:z
5289:+
5286:=
5277:3
5273:x
5265:,
5258:+
5254:z
5245:=
5236:2
5232:x
5224:,
5217:+
5213:z
5207:+
5204:=
5195:1
5191:x
5173:)
5171:x
5169:(
5167:Q
5162:)
5160:z
5158:(
5156:q
5150:−
5147:z
5141:+
5138:z
5132:0
5129:a
5125:z
5122:2
5119:a
5115:z
5112:4
5109:a
5105:z
5103:(
5101:q
5095:z
5089:q
5081:)
5079:x
5077:(
5075:Q
5069:x
5065:z
5034:0
5030:a
5026:+
5021:2
5017:x
5011:2
5007:a
5003:+
4998:4
4994:x
4988:4
4984:a
4980:=
4977:)
4974:x
4971:(
4968:Q
4953:1
4950:a
4946:3
4943:a
4914:)
4912:x
4910:(
4908:Q
4902:Q
4889:R
4879:)
4877:x
4875:(
4873:Q
4855:)
4853:x
4851:(
4849:Q
4831:.
4828:)
4823:0
4819:d
4815:+
4812:x
4807:1
4803:d
4799:+
4794:2
4790:x
4784:2
4780:d
4776:(
4773:)
4768:0
4764:c
4760:+
4757:x
4752:1
4748:c
4744:+
4739:2
4735:x
4729:2
4725:c
4721:(
4718:=
4715:)
4712:x
4709:(
4706:Q
4680:)
4675:0
4671:b
4667:+
4664:x
4659:1
4655:b
4651:+
4646:2
4642:x
4636:2
4632:b
4628:+
4623:3
4619:x
4613:3
4609:b
4605:(
4602:)
4597:1
4593:x
4586:x
4583:(
4580:=
4577:)
4574:x
4571:(
4568:Q
4555:)
4553:x
4551:(
4549:Q
4536:)
4534:x
4532:(
4530:S
4525:)
4523:x
4521:(
4519:R
4514:)
4512:x
4510:(
4508:S
4504:x
4502:(
4500:R
4496:x
4494:(
4492:Q
4470:.
4465:0
4461:a
4457:+
4454:x
4449:1
4445:a
4441:+
4436:2
4432:x
4426:2
4422:a
4418:+
4413:3
4409:x
4403:3
4399:a
4395:+
4390:4
4386:x
4380:4
4376:a
4372:=
4369:)
4366:x
4363:(
4360:Q
4319:,
4316:0
4308:0
4283:0
4280:=
4255:.
4252:a
4248:/
4244:b
4238:=
4233:0
4229:x
4225:3
4222:+
4217:1
4213:x
4190:1
4186:x
4161:;
4158:)
4155:c
4152:+
4149:x
4146:b
4143:3
4140:+
4135:2
4131:x
4127:a
4124:6
4121:(
4118:2
4096:0
4092:x
4067:,
4064:0
4061:=
4056:1
4031:,
4028:0
4025:=
4020:0
3995:0
3992:=
3978:.
3958:q
3934:.
3929:4
3924:)
3916:a
3913:4
3909:b
3903:+
3900:x
3896:(
3868:S
3848:Q
3828:,
3825:0
3822:=
3819:S
3797:.
3792:1
3767:,
3762:1
3733:2
3728:1
3701:,
3698:0
3692:Q
3668:2
3663:1
3653:=
3646:3
3641:0
3633:4
3625:2
3620:1
3593:,
3590:0
3587:=
3582:0
3557:0
3525:.
3521:)
3511:3
3506:0
3496:2
3490:1
3480:(
3470:=
3437:3
3419:0
3406:a
3403:3
3399:2
3394:+
3391:p
3383:3
3380:2
3368:2
3365:1
3360:=
3357:S
3323:;
3320:Q
3300:S
3280:Q
3260:,
3257:0
3233:.
3227:Q
3189:,
3174:=
3169:3
3164:0
3156:4
3148:2
3143:1
3108:e
3105:c
3102:a
3091:2
3087:d
3083:a
3077:+
3074:e
3069:2
3065:b
3058:+
3055:d
3052:c
3049:b
3046:9
3038:3
3034:c
3030:2
3027:=
3018:1
3006:e
3003:a
2997:+
2994:d
2991:b
2988:3
2980:2
2976:c
2972:=
2963:0
2932:Q
2925:S
2900:3
2895:2
2887:3
2882:0
2874:4
2866:2
2861:1
2851:+
2846:1
2834:=
2827:Q
2817:)
2811:Q
2806:0
2796:+
2793:Q
2789:(
2782:a
2779:3
2775:1
2770:+
2767:p
2759:3
2756:2
2744:2
2741:1
2736:=
2729:S
2688:3
2684:a
2680:8
2675:d
2670:2
2666:a
2662:8
2659:+
2656:c
2653:b
2650:a
2647:4
2639:3
2635:b
2628:=
2621:q
2609:2
2605:a
2601:8
2594:2
2590:b
2586:3
2580:c
2577:a
2574:8
2568:=
2561:p
2541:q
2537:p
2512:S
2509:q
2501:p
2498:2
2490:2
2486:S
2482:4
2472:2
2469:1
2461:S
2458:+
2452:a
2449:4
2445:b
2437:=
2425:4
2422:,
2419:3
2415:x
2403:S
2400:q
2395:+
2392:p
2389:2
2381:2
2377:S
2373:4
2363:2
2360:1
2352:S
2343:a
2340:4
2336:b
2328:=
2316:2
2313:,
2310:1
2306:x
2272:a
2254:0
2251:=
2248:e
2245:+
2242:x
2239:d
2236:+
2231:2
2227:x
2223:c
2220:+
2215:3
2211:x
2207:b
2204:+
2199:4
2195:x
2191:a
2177:4
2174:x
2168:3
2165:x
2159:2
2156:x
2150:1
2147:x
2128:0
2125:=
2122:d
2119:+
2116:x
2113:c
2110:+
2105:2
2101:x
2097:b
2094:+
2089:3
2085:x
2081:a
2078:+
2073:4
2069:x
2036:;
2031:2
2027:P
2023:+
2020:D
2017:3
2014:=
2009:0
1999:2
1995:a
1981:D
1977:P
1971:0
1969:∆
1965:D
1961:P
1955:0
1953:∆
1937:a
1935:4
1932:/
1928:b
1922:−
1916:0
1914:∆
1907:R
1903:P
1896:P
1889:D
1882:D
1876:0
1874:∆
1867:R
1863:D
1859:P
1855:D
1846:0
1844:∆
1840:D
1836:P
1816:D
1812:P
1805:D
1801:P
1757:4
1753:b
1749:3
1743:d
1740:b
1735:2
1731:a
1721:c
1716:2
1712:b
1708:a
1702:+
1697:2
1693:c
1687:2
1683:a
1673:e
1668:3
1664:a
1657:=
1654:D
1628:,
1625:e
1622:a
1616:+
1613:d
1610:b
1607:3
1599:2
1595:c
1591:=
1586:0
1563:a
1561:8
1558:/
1554:R
1532:,
1529:c
1526:b
1523:a
1520:4
1512:2
1508:a
1504:d
1501:8
1498:+
1493:3
1489:b
1485:=
1482:R
1460:a
1458:8
1455:/
1451:P
1427:2
1423:b
1419:3
1413:c
1410:a
1407:8
1404:=
1401:P
1369:2
1365:d
1359:2
1355:c
1349:2
1345:b
1341:+
1338:e
1333:3
1329:c
1323:2
1319:b
1315:4
1307:3
1303:d
1297:3
1293:b
1289:4
1283:e
1280:d
1277:c
1272:3
1268:b
1261:+
1256:2
1252:e
1246:4
1242:b
1230:2
1226:d
1220:3
1216:c
1212:a
1209:4
1196:e
1191:4
1187:c
1183:a
1177:+
1172:3
1168:d
1164:c
1161:b
1158:a
1152:+
1149:e
1146:d
1141:2
1137:c
1133:b
1130:a
1121:e
1116:2
1112:d
1106:2
1102:b
1098:a
1095:6
1087:2
1083:e
1079:c
1074:2
1070:b
1066:a
1060:+
1048:4
1044:d
1038:2
1034:a
1024:e
1019:2
1015:d
1011:c
1006:2
1002:a
995:+
990:2
986:e
980:2
976:c
970:2
966:a
954:2
950:e
946:d
943:b
938:2
934:a
922:3
918:e
912:3
908:a
897:=
872:a
854:0
851:=
848:e
845:+
842:x
839:d
836:+
831:2
827:x
823:c
820:+
815:3
811:x
807:b
804:+
799:4
795:x
791:a
751:.
748:)
740:(
733:=
728:2
722:5
717:+
714:1
708:=
702:H
699:G
694:G
691:F
667:G
663:F
659:G
648:H
640:G
636:F
558:z
447:a
439:a
418:.
415:e
412:+
407:2
403:x
399:c
396:+
391:4
387:x
383:a
380:=
377:)
374:x
371:(
368:f
326:a
308:,
305:0
302:=
299:e
296:+
293:x
290:d
287:+
282:2
278:x
274:c
271:+
266:3
262:x
258:b
255:+
250:4
246:x
242:a
208:a
191:,
188:e
185:+
182:x
179:d
176:+
171:2
167:x
163:c
160:+
155:3
151:x
147:b
144:+
139:4
135:x
131:a
128:=
125:)
122:x
119:(
116:f
87:x
83:x
79:x
71:x
59:x
37:.
30:.
23:.
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