# Cubic Equation Calculator

Solve any cubic equation ax³ + bx² + cx + d = 0 instantly. Finds all three roots (real and complex), discriminant, and root type. Free online cubic solver.

## What this calculates

Solve any cubic equation of the form ax³ + bx² + cx + d = 0. This calculator finds all three roots, whether they are real or complex, computes the discriminant, and identifies the root type.

## Inputs

- **Coefficient a (x³ term)** — The coefficient of x³. Must not be zero.
- **Coefficient b (x² term)** — The coefficient of x².
- **Coefficient c (x term)** — The coefficient of x.
- **Coefficient d (constant term)** — The constant term.

## Outputs

- **Discriminant** — Determines the nature and number of roots.
- **Root Type** — formatted as text — Whether the roots are three real, one real + two complex, or repeated.
- **Root 1 (x₁)** — formatted as text — The first root of the cubic equation.
- **Root 2 (x₂)** — formatted as text — The second root of the cubic equation.
- **Root 3 (x₃)** — formatted as text — The third root of the cubic equation.

## Details

A cubic equation is a third-degree polynomial equation. Unlike quadratics, cubics always have at least one real root. The general solution uses Cardano's method, published in 1545.

**The Discriminant:**

The discriminant D = -4p³ - 27q² (where p and q come from the depressed cubic) determines the root types:

- **D > 0:** Three distinct real roots
- **D = 0:** A repeated root (one double root plus one single root, or a triple root)
- **D < 0:** One real root and two complex conjugate roots

**Depressed Cubic:**

The substitution x = t - b/(3a) eliminates the x² term, converting ax³ + bx² + cx + d = 0 into the simpler form t³ + pt + q = 0. This depressed cubic is solved using Cardano's formula or, when there are three real roots, the trigonometric method.

**Applications:**

Cubic equations appear in engineering (beam deflection), physics (equations of state for gases), economics (cost optimization), and computer graphics (Bezier curves). They are also the simplest polynomials where complex roots first become unavoidable in the solution process.

## Frequently Asked Questions

**Q: How many roots does a cubic equation have?**

A: A cubic equation always has exactly three roots (counting multiplicity), which follow from the Fundamental Theorem of Algebra. These can be three distinct real roots, one real root plus two complex conjugate roots, or a combination involving repeated roots (like a double root plus a single root, or a triple root).

**Q: What is Cardano's formula?**

A: Cardano's formula solves depressed cubics of the form t³ + pt + q = 0. You compute u = cbrt(-q/2 + sqrt(q²/4 + p³/27)) and v = cbrt(-q/2 - sqrt(q²/4 + p³/27)), then t = u + v. Published by Gerolamo Cardano in 1545, it was one of the first formulas for solving polynomials beyond degree 2.

**Q: What is a depressed cubic?**

A: A depressed cubic has no x² term: t³ + pt + q = 0. Any cubic ax³ + bx² + cx + d = 0 can be converted to this form by substituting x = t - b/(3a). This simplification makes the equation easier to solve with Cardano's formula or the trigonometric method.

**Q: Can a cubic equation have all complex roots?**

A: No. Every cubic with real coefficients has at least one real root. This is because complex roots of polynomials with real coefficients always come in conjugate pairs. Since three is odd, at least one root must be real. The remaining two can be either both real or a complex conjugate pair.

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Source: https://vastcalc.com/calculators/math/cubic-equation
Category: Math
Last updated: 2026-04-08