Use this calculator to solve polynomial equations with an order of 3 such as ax 3 + bx 2 + cx + d = 0 for x including complex solutions. Enter positive or negative values for a, b, c and d and the calculator will find all solutions for x. Enter 0 if that term is not present in your cubic equation. There are either one or three possible real root solutions for x for any cubic equation. You may have only two distinct solutions as in the case x = 1, x = 5, x = 5, however there are still three real roots.
A cubic equation is an algebraic equation with a degree of 3. This means that the highest exponent in the equation is 3. Written in standard form, where a ≠ 0 a cubic equation looks like this: \[ ax^3 + bx^2 + cx + d = 0 \] The b, c or d terms may be missing from the equation, or the a term might be 1. As long as there is an ax 3 value you have a cubic equation.
There are multiple ways to solve cubic equations. The method you use depends on your equation. Check the guidance below for the best way to solve your cubic equation.
If your equation has a constant d use these methods to solve the cubic equation:
Vieta's formulas show the relationship between the coefficients of a polynomial and the sums and products of its roots. If you know one root, you may be able to do substitutions and figure out the others.
For a cubic equation ax 3 + bx 2 + cx + d = 0, let p, q and r be the 3 roots of the equation. So:
\[ (x-p)(x-q)(x-r) = 0 \text <, just as >\] \[ ax^3 + bx^2 + cx + d = 0 \]Vieta's Formulas use these equivalences to show how the roots relate to the coefficients of the cubic equation. The equivalences are listed below, along with the proof.