Cubic Equation Solver – ax³ + bx² + cx + d = 0

A cubic equation is a polynomial equation of degree three, written as ax³ + bx² + cx + d = 0, where a ≠ 0. It always has three roots (real or complex) due to the Fundamental Theorem of Algebra.

Cardano's method reduces the general cubic to a depressed cubic (without x² term) via substitution. The roots are then expressed using cube roots of complex numbers, handled through the discriminant Δ. Our solver implements the complete algebraic solution, also covering trigonometric cases for three real roots.

The discriminant Δ = (q/2)² + (p/3)³, derived from the depressed cubic t³ + pt + q = 0. If Δ > 0, there is one real root and two complex conjugate roots. Δ = 0 indicates multiple real roots (at least two equal). Δ < 0 gives three distinct real roots.

Complex roots appear in conjugate pairs, formatted as a + bi and a – bi. Our tool clearly labels real and imaginary parts, using standard mathematical notation. All values are approximated to 6 decimal places for readability.

No. If a = 0, the equation becomes quadratic or linear. This tool is designed specifically for cubic equations. For quadratics, please use a dedicated quadratic solver.

The solver uses floating‑point arithmetic and may round to 6 decimal places. Extremely large or small coefficients can introduce small numerical errors, but the algorithm is stable for typical inputs.