What is a cubic equation?

A cubic equation is a polynomial equation of degree 3, written as ax³ + bx² + cx + d = 0, where a ≠ 0. Every cubic has exactly three roots (counting multiplicity) in the complex numbers.

Can a cubic equation have complex roots?

Yes. A cubic always has at least one real root. The other two can be real or a conjugate pair of complex numbers (e.g., 2 + 3i and 2 − 3i).

What method does this solver use?

The solver uses the discriminant of the depressed cubic. When all three roots are real and distinct (discriminant > 0), the trigonometric (Viete) method is used for numerical stability. Otherwise Cardano's formula is applied.

Setting a = 0 reduces the equation to a quadratic. The solver will display an error and prompt you to use a quadratic solver instead.

Are the results exact or approximate?

Results are computed in 64-bit floating-point arithmetic (JavaScript's standard). For most practical purposes the answers are accurate to at least 10 significant digits.

How do I interpret complex roots?

Complex roots are shown in a + bi format. For example, "1.5 + 2.598i" means the root is 1.5 + 2.598i. Complex roots always appear in conjugate pairs for polynomials with real coefficients.

Cubic Equation Solver

Enter coefficients a, b, c, d for ax³ + bx² + cx + d = 0 and instantly see all three roots (real or complex).

Enter Coefficients

Equation: ax³ + bx² + cx + d = 0

a (x³ coefficient)

b (x² coefficient)

c (x coefficient)

d (constant)

Preview

x³ − 3x + 2 = 0

Enter coefficients and click Solve to find the roots.

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Summary

This cubic equation solver computes all three roots of any cubic polynomial ax³ + bx² + cx + d = 0. It uses the discriminant method to classify the root types and applies Cardano's formula or trigonometric approach for accurate results. Both real and complex roots are displayed in standard mathematical notation.

How it works

Enter the coefficients a, b, c, and d in the input fields.
The solver normalizes the equation by dividing through by a.
The discriminant is computed to determine how many real roots exist.
Cardano's formula or the trigonometric method is applied based on the discriminant.
All three roots are displayed, with complex roots shown in a + bi notation.

Use cases

Verify hand-calculated roots of a cubic polynomial.
Solve physics and engineering equations that produce cubic forms.
Check roots during calculus or algebra coursework.
Find the turning points of cubic functions in graphing problems.
Solve cubic volume or optimization problems quickly.
Test understanding of complex number arithmetic with known cubics.

Frequently Asked Questions

Last updated: 2026-05-29 · Reviewed by Nham Vu