Cubic equation

[6] The problem of doubling the cube involves the simplest and oldest studied cubic equation, and one for which the ancient Egyptians did not believe a solution existed.

[7] In the 5th century BC, Hippocrates reduced this problem to that of finding two mean proportionals between one line and another of twice its length, but could not solve this with a compass and straightedge construction,[8] a task which is now known to be impossible.

[3][9] Hippocrates, Menaechmus and Archimedes are believed to have come close to solving the problem of doubling the cube using intersecting conic sections,[8] though historians such as Reviel Netz dispute whether the Greeks were thinking about cubic equations or just problems that can lead to cubic equations.

[10] In the 7th century, the Tang dynasty astronomer mathematician Wang Xiaotong in his mathematical treatise titled Jigu Suanjing systematically established and solved numerically 25 cubic equations of the form x3 + px2 + qx = N, 23 of them with p, q ≠ 0, and two of them with q = 0.

In the 11th century, the Persian poet-mathematician, Omar Khayyam (1048–1131), made significant progress in the theory of cubic equations.

In an early paper, he discovered that a cubic equation can have more than one solution and stated that it cannot be solved using compass and straightedge constructions.

[12][a] In his later work, the Treatise on Demonstration of Problems of Algebra, he wrote a complete classification of cubic equations with general geometric solutions found by means of intersecting conic sections.

It may be, however, that men who come after us will succeed.”[15]In the 12th century, the Indian mathematician Bhaskara II attempted the solution of cubic equations without general success.

He also used the concepts of maxima and minima of curves in order to solve cubic equations which may not have positive solutions.

[18] In his book Flos, Leonardo de Pisa, also known as Fibonacci (1170–1250), was able to closely approximate the positive solution to the cubic equation x3 + 2x2 + 10x = 20.

[19] In the early 16th century, the Italian mathematician Scipione del Ferro (1465–1526) found a method for solving a class of cubic equations, namely those of the form x3 + mx = n. In fact, all cubic equations can be reduced to this form if one allows m and n to be negative, but negative numbers were not known to him at that time.

Del Ferro kept his achievement secret until just before his death, when he told his student Antonio Fior about it.

In 1535, Niccolò Tartaglia (1500–1557) received two problems in cubic equations from Zuanne da Coi and announced that he could solve them.

Fior received questions in the form x3 + mx2 = n, which proved to be too difficult for him to solve, and Tartaglia won the contest.

Some years later, Cardano learned about del Ferro's prior work and published del Ferro's method in his book Ars Magna in 1545, meaning Cardano gave Tartaglia six years to publish his results (with credit given to Tartaglia for an independent solution).

[20] Cardano noticed that Tartaglia's method sometimes required him to extract the square root of a negative number.

Gerolamo Cardano is credited with publishing the first formula for solving cubic equations, attributing it to Scipione del Ferro and Niccolo Fontana Tartaglia.

Also, the use of principal cube root may give a wrong result if the coefficients are non-real complex numbers.

Nevertheless, purely real expressions of the solutions may be obtained using trigonometric functions, specifically in terms of cosines and arccosines.

If p ≠ 0 and the inequalities on the right are not satisfied (the case of three real roots), the formulas remain valid but involve complex quantities.

This implies that the old problems of angle trisection and doubling the cube, set by ancient Greek mathematicians, cannot be solved by compass-and-straightedge construction.

Viète's trigonometric expression of the roots in the three-real-roots case lends itself to a geometric interpretation in terms of a circle.

When the graph of a cubic function is plotted in the Cartesian plane, if there is only one real root, it is the abscissa (x-coordinate) of the horizontal intercept of the curve (point R on the figure).

Marden's theorem says that the points representing the roots of the derivative of the cubic are the foci of the Steiner inellipse of the triangle—the unique ellipse that is tangent to the triangle at the midpoints of its sides.

If that angle is greater than ⁠π/3⁠, the major axis is vertical and its foci, the roots of the derivative, are complex conjugates.

This method is due to Scipione del Ferro and Tartaglia, but is named after Gerolamo Cardano who first published it in his book Ars Magna (1545).

Vieta's substitution is a method introduced by François Viète (Vieta is his Latin name) in a text published posthumously in 1615, which provides directly the second formula of § Cardano's method, and avoids the problem of computing two different cube roots.

In his paper Réflexions sur la résolution algébrique des équations ("Thoughts on the algebraic solving of equations"),[36] Joseph Louis Lagrange introduced a new method to solve equations of low degree in a uniform way, with the hope that he could generalize it for higher degrees.

[37][38][39] Apart from the fact that nobody had previously succeeded, this was the first indication of the non-existence of an algebraic formula for degrees 5 and higher; as was later proved by the Abel–Ruffini theorem.

Using Newton's identities, it is straightforward to express them in terms of the elementary symmetric functions of the roots, giving