Cubic equation

In mathematics, and more specifically algebra, a cubic equation is an equation involving only polynomials of the third degree. Although cubic equations occur less frequently in real-world problems than quadratic or linear equations, they still do occur.

Every cubic equation with real number coefficients has at most three real roots, as dictated by the fundamental theorem of algebra. Solving cubic equations is more difficult than solving quadratic equations, where the quadratic formula can be applied, or linear equations, which can be solved using arithmetic operations. As with quadratic equations, there is a closed formula for solving cubic equations, the cubic formula.

The cubic formula is much more complicated than the quadratic formula. Furthermore, there are examples of cubic equations with real coefficients and three distinct real solutions for which the cubic formula requires one to calculate a root of a complex number. This is more difficult than finding the root of a real number, which is all that the quadratic formula requires. It is precisely this difficulty which first demonstrated to mathematicians the utility of complex numbers.

Because the cubic formula is unwieldy, most students are taught to solve only small class of cubic equations, namely, those with at least one rational root. For such equations, all roots can be found by factoring and a possible application of the quadratic formula. When the coefficients are rational, the initial factor can easily be found using the rational root theorem.

Solutions of cubic equations

Solving a cubic equation begins, if necessary, by simplifying the polynomials. Using the elementary arithmetic operations and combining like terms, every cubic equation in one variable can be put in the form

$x^{3}+ax^{2}+bx+c=0$.

The equation has at least one real root. This is a consequence of the intermediate value theorem and the observation that as $x$ moves to the right or left on the real line, so does the value of the polynomial on the left of the above equation.

The discriminant

Just as quadratic polynomials do, each cubic polynomial has a discriminant which is related to the roots of the polynomial. The discriminant of the above polynomial is

$\Delta =a^{2}b^{2}-4b^{3}-4a^{3}c-27c^{2}+18abc$.

As with quadratic equations, the sign of the discriminant of a cubic equation with real coefficients determines three possible qualitative descriptions of the roots:

• If $\Delta >0$, then the equation has three distinct real roots.
• If $\Delta <0$, then the equation has two complex conjugate nonreal roots and one real root.
• If $\Delta =0$, then the equation has a repeated root. It can have either a real root of multiplicity three, or it can have a real root of multiplicity two and another real root.