Cubic equation

In algebra, a cubic equation in one variable is an equation of the form

Graph of a cubic function with 3 real roots (where the curve crosses the horizontal axis at y = 0). The case shown has two critical points. Here the function is f(x) = (x3 + 3x2 − 6x − 8)/4.

in which a is nonzero.

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:

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.