🧮 Cubic Equation Calculator – Solve & Understand

The Cubic Equation Calculator helps you quickly and accurately solve any cubic equation of the form:

ax³ + bx² + cx + d = 0

Just enter the values for a, b, c, and d, choose the number of decimal places you want (precision), then click “Solve”. You’ll instantly get the 3 roots (x₁, x₂, x₃) along with a step-by-step explanation.

📘 How to Solve a Cubic Equation – Step-by-Step Guide

ax³ + bx² + cx + d = 0

where $a\mathrm{\ne }0$

Step 1: Normalize the Equation

Divide the entire equation by $a$ to simplify:

$$x^3+\frac{b}{a}{x}^2+\frac{c}{a}x+\frac{d}{a}=0$$

Let:

$$A=\frac{b}{a},B=\frac{c}{a},C=\frac{d}{a}$$

Step 2: Depress the Cubic (Remove the $x^2$ term)

Make a substitution to eliminate the quadratic term:

$$x=t-\frac{A}{3}$$

This transforms the equation into a depressed cubic form:

$$t^3+pt+q=0$$

where

$$p=B-\frac{A^2}{3},q=\frac{2A^3}{27}-\frac{AB}{3}+C$$

Step 3: Calculate the Discriminant

Calculate the discriminant $\Delta$:

$$\mathrm{\Delta }={\left(\frac{q}{2}\right)}^2+{\left(\frac{p}{3}\right)}^3$$

• If $\mathrm{\Delta }$: One real root and two complex conjugate roots.

• If $\mathrm{\Delta }$: All roots are real, and at least two are equal.

• If $\mathrm{\Delta }$: All three roots are real and distinct.

Step 4: Solve for Roots Using Cardano’s Formula

Define:

$$u=\sqrt[3]{-\frac{q}{2}+\sqrt{\mathrm{\Delta }}},v=\sqrt[3]{-\frac{q}{2}-\sqrt{\mathrm{\Delta }}}$$

Then,

$$t=u+v$$

Recall:

$$x=t-\frac{A}{3}$$

This gives the real root. For complex or multiple roots, different formulas or trigonometric methods are applied depending on $\Delta$.

Step 5: For Three Real Roots (When $\Delta < 0$)

Use the trigonometric solution:

$$t_k=2\sqrt{-\frac{p}{3}}\cos \left(\frac{\theta +2k\pi }{3}\right),k=0,1,2$$

where

$$\cos \theta =-\frac{q}{2}/\sqrt{-{\left(\frac{p}{3}\right)}^3}$$

Then,

$$x_k=t_k-\frac{A}{3}$$

Summary

1. Normalize coefficients.

2. Remove the $x^2$ term by substitution.

3. Calculate $p$, $q$, and discriminant $\Delta$.

4. Apply Cardano’s formula or trigonometric formulas depending on $\Delta$.

5. Find roots $x$.