Cubic Equation Calculator

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\[ ax^3 + bx^2 + cx + d = 0 \]

a =

b =

c =

d =

Answer:

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Calculator Use

Use this calculator to solve polynomial equations with an order of 3 such as ax³ + bx² + 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.

What is a Cubic Equation?

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 ax³ value you have a cubic equation.

How to Solve 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.

Methods to Solve Cubic Equations That Do Not Have a Constant, d

If your equation does not have a constant d you can factor out the x, so one of your answers is x = 0. Then you can use one of these methods to solve the resulting quadratic equation, which is simply an equation of degree 2:

Factor the resulting quadratic equation
Solve the resulting quadratic equation by Completing the Square
Solve the resulting quadratic equation with the Quadratic Formula

Methods to Solve Cubic Equations That Have a Constant, d

If your equation has a constant d use these methods to solve the cubic equation:

Factoring the cubic equation
Using Vieta's Formulas, described below

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 ax³ + bx² + 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.

Vieta's Equivalents

Root Expression

Equals

\[ p + q + r \]

\[ - \dfrac{b}{a} \]

\[ pq + qr + rp \]

\[ \dfrac{c}{a} \]

\[ pqr \]

\[ - \dfrac{d}{a} \]

You can follow the steps below to see how these equivalents are derived from the cubic equation expressions. Remember, standard form of a cubic equation is:

\[ ax^3 + bx^2 + cx + d = 0 \]

If you divide each side of the equation by a you get:

\[ x^3 + \dfrac{b}{a}x^2 + \dfrac{c}{a}x + \dfrac{d}{a} = 0 \]

Switching gears, take the root expressions and multiply them out:

\[ (x-p)(x-q)(x-r) = 0 \]

\[ (x−p)(x^2−qx−rx+qr) = 0 \]

\[ x^3 − qx^2 - rx^2 + qrx - px^2 \]

\[ + pqx + prx − pqr = 0 \]

\[ x^3 + [-(p+q+r)]x^2 \]

\[ + (pq+qr+pr)x + (−pqr) = 0 \]

Compare this last equation to the reduced cubic equation below. Note the coefficients for x³, x², x and the constant above:

\[ x^3 + \dfrac{b}{a}x^2 + \dfrac{c}{a}x + \dfrac{d}{a} = 0 \]

You can see these patterns,

\[ p + q + r = - \dfrac{b}{a} \]

\[ pq + qr + rp = \dfrac{c}{a} \]

\[ pqr = \dfrac{d}{a} \]

References

Wikipedia, Vieta's Formulas, accessed April 14, 2023.

Brilliant.org, Cubic Equations, accessed April 14, 2023.