What is cubic equation for?

A cubic equation is a third-degree polynomial equation that can be written in the standard form \(ax^{3}+bx^{2}+cx+d=0\), where \(a\ne 0\). Cubic equations have three roots (solutions), which can be real or complex, and are sometimes equal. Solutions can be found by factoring, using the Rational Root Theorem, or by using the cubic formula, which is complex but provides an exact solution.

The solutions of this equation are called roots of the cubic function defined by the left-hand side of the equation. If all of the coefficients a, b, c, and d of the cubic equation are real numbers, then it has at least one real root (this is true for all odd-degree polynomial functions). All of the roots of the cubic equation can be found by the following means:

algebraically: more precisely, they can be expressed by a cubic formula involving the four coefficients, the four basic arithmetic operations, square roots, and cube roots. (This is also true of quadratic (second-degree) and quartic (fourth-degree) equations, but not for higher-degree equations, by the Abel–Ruffini theorem.)
geometrically: using Omar Kahyyam's method.
trigonometrically
numerical approximations of the roots can be found using root-finding algorithms such as Newton's method.
The coefficients do not need to be real numbers. Much of what is covered below is valid for coefficients in any field with characteristic other than 2 and 3. The solutions of the cubic equation do not necessarily belong to the same field as the coefficients. For example, some cubic equations with rational coefficients have roots that are irrational (and even non-real) complex numbers.