Cubic equation

If ${\displaystyle \Delta >0,}$ the cubic has three distinct real

If ${\displaystyle \Delta <0,}$ the cubic has one real root and two non-real roots.

and doubling the cube are two ancient problems of geometry that have been proved to not be solvable by straightedge and compass construction, because they are equivalent to solving a cubic equation.

states that the foci of the Steiner inellipse of any triangle can be found by using the cubic function whose roots are the coordinates in the complex plane of the triangle's three vertices. The roots of the first derivative of this cubic are the complex coordinates of those foci.

The of a regular heptagon can be expressed in terms of the roots of a cubic. Further, the ratios of the long diagonal to the side, the side to the short diagonal, and the negative of the short diagonal to the long diagonal all satisfy a particular cubic equation. In addition, the ratio of the inradius to the circumradius of a heptagonal triangle is one of the solutions of a cubic equation. The values of trigonometric functions of angles related to ${\displaystyle 2\pi /7}$ satisfy cubic equations.

Given the cosine (or other trigonometric function) of an arbitrary angle, the cosine of is one of the roots of a cubic.

The solution of the general relies on the solution of its resolvent cubic.

The of a 3×3 matrix are the roots of a cubic polynomial which is the characteristic polynomial of the matrix.

The of a third-order constant coefficients or Cauchy–Euler (equidimensional variable coefficients) linear differential equation or difference equation is a cubic equation.

Intersection points of cubic and straight line can be computed using direct cubic equation representing Bézier curve.

of a quartic function are found by solving a cubic equation (the derivative set equal to zero).

of a quintic function are the solution of a cubic equation (the second derivative set equal to zero).

Guilbeau, Lucye (1930), "The History of the Solution of the Cubic Equation", Mathematics News Letter, 5 (4): 8–12, :10.2307/3027812, JSTOR 3027812

Anglin, W. S.; Lambek, Joachim (1995), , The Heritage of Thales, Springers, pp. 125–131, ISBN 978-0-387-94544-6 Ch. 24.

Dence, T. (November 1997), "Cubics, chaos and Newton's method", Mathematical Gazette, 81 (492), : 403–408, doi:10.2307/3619617, ISSN 0025-5572, JSTOR 3619617, S2CID 125196796

Dunnett, R. (November 1994), "Newton–Raphson and the cubic", Mathematical Gazette, 78 (483), : 347–348, doi:10.2307/3620218, ISSN 0025-5572, JSTOR 3620218, S2CID 125643035

(2009), Basic algebra, vol. 1 (2nd ed.), Dover, ISBN 978-0-486-47189-1

Mitchell, D. W. (November 2007), "Solving cubics by solving triangles", Mathematical Gazette, 91, : 514–516, doi:10.1017/S0025557200182178, ISSN 0025-5572, S2CID 124710259

Mitchell, D. W. (November 2009), "Powers of φ as roots of cubics", Mathematical Gazette, 93, , doi:10.1017/S0025557200185237, ISSN 0025-5572, S2CID 126286653

Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. (2007), , Numerical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press, ISBN 978-0-521-88068-8

Rechtschaffen, Edgar (July 2008), "Real roots of cubics: Explicit formula for quasi-solutions", Mathematical Gazette, 92, : 268–276, doi:10.1017/S0025557200183147, ISSN 0025-5572, S2CID 125870578

Zucker, I. J. (July 2008), "The cubic equation – a new look at the irreducible case", Mathematical Gazette, 92, : 264–268, doi:10.1017/S0025557200183135, ISSN 0025-5572, S2CID 125986006

, Encyclopedia of Mathematics, EMS Press, 2001 [1994]

on MacTutor archive.

– YouTube video by Mathologer about the history of cubic equations and Cardano's solution, as well as Ferrari's solution to quartic equations