Roots Calculator Cubic

Find the real and complex roots of cubic equations with our roots calculator. Solve equations of the form x³ + ax² + bx + c = 0 with step-by-step solutions and formula explanations.

How to Use This Calculator

Enter the coefficients of your cubic equation in the form x³ + ax² + bx + c = 0. The calculator will find all roots (real and complex) of the equation.

For equations without an x² term (a = 0), the calculator will use the simplified cubic formula. For equations with a = 0 and b = 0, it will use the simple cubic root formula.

Cubic Roots Formula

The general solution for a cubic equation x³ + ax² + bx + c = 0 can be found using Cardano's formula:

x = -a/3 + (∛[(q/2) + √(q²/4 + p³/27)] + ∛[(q/2) - √(q²/4 + p³/27)])/3

where p = b - a²/3 and q = c - ab/3 + 2a³/27

For equations where a = 0 (x³ + bx + c = 0), the simplified formula is:

x = ∛[-q/2 + √(q²/4 + p³/27)] + ∛[-q/2 - √(q²/4 + p³/27)]

where p = b and q = c

For equations where a = 0 and b = 0 (x³ + c = 0), the solution is simply:

x = ∛(-c)

Real-World Examples

Example 1: Simple Cubic Equation

Solve x³ - 6x² + 11x - 6 = 0

Using the calculator, we find the roots are x = 1, x = 2, and x = 3.

Example 2: Complex Roots

Solve x³ + x + 1 = 0

The calculator finds one real root and two complex roots:

x ≈ -1.3247
x ≈ 0.6624 + 0.5623i
x ≈ 0.6624 - 0.5623i

Example 3: Missing Quadratic Term

Solve x³ + 5x + 6 = 0

The calculator uses the simplified formula and finds the roots x = -2 and x = -3 (with multiplicity 1).

Frequently Asked Questions

What is a cubic equation?: A cubic equation is a polynomial equation of degree 3, typically in the form x³ + ax² + bx + c = 0.
How many roots can a cubic equation have?: A cubic equation always has three roots, which can be real or complex. Some roots may be repeated.
What are complex roots?: Complex roots are solutions to equations that involve imaginary numbers (i, where i² = -1). They appear as pairs of conjugate roots.
Can this calculator solve quartic equations?: No, this calculator is specifically designed for cubic equations. For quartic equations, you would need a different calculator.
What if my equation doesn't have an x² term?: The calculator automatically uses the simplified formula when a = 0, so you don't need to adjust your input.