Cubic Equation Solver

What does this calculator do?

Enter the four coefficients a, b, c, d of a cubic equation ax³ + bx² + cx + d = 0 and this tool solves for all three roots of x — real or complex — using Cardano's method, with a step-by-step derivation shown alongside the answer.

How to use it

1. Enter a (the coefficient of x³ — must not be zero), b (coefficient of x²), c (coefficient of x), and d (the constant term).
2. Read the three roots from the Roots field, the discriminant from Discriminant (Δ), and the full derivation from Step-by-Step Solution.

The math

Dividing the equation by a gives x³ + Bx² + Cx + D = 0. Substituting x = t − B/3 eliminates the quadratic term, leaving a depressed cubic:

t³ + pt + q = 0
p = C − B²/3
q = 2B³/27 − BC/3 + D

The sign of the discriminant Δ = (q/2)² + (p/3)³ decides the shape of the roots:

• Δ > 0 — one real root and a complex-conjugate pair, found directly from Cardano's radical formula: u = ∛(−q/2 + √Δ), v = ∛(−q/2 − √Δ), with the real root t = u + v.
• Δ = 0 — all three roots are real, with at least two equal: a triple root at t = 0 when p and q are both zero, otherwise a simple root and a double root.
• Δ < 0 — three distinct real roots. Cardano's radical formula would require taking a complex cube root of a real number here (the historical "casus irreducibilis"), so this calculator uses the equivalent trigonometric form instead, which stays entirely in real arithmetic.

Finally, x = t − B/3 undoes the substitution to recover the roots of the original equation.

Example use cases

• Algebra homework and coursework — checking cubic factoring or root-finding work, including cases that don't factor into nice integers.
• Engineering and physics — cubic equations show up in beam deflection, fluid dynamics (van der Waals equation of state), and control-system characteristic equations.
• Verifying a known factorization — enter the expanded coefficients of a cubic you've factored by hand and confirm the roots match.

Privacy

All calculations run locally in your browser — your input is never sent to a server.

Related tools

For quadratic equations, see the Quadratic Formula Solver. For simple linear equations, see the Linear Equation Solver.