Cubic equation solver

A cubic equation is any polynomial equation of degree three written as a times x cubed plus b times x squared plus c times x plus d equals zero when the leading coefficient a is not zero. Cardano's classical closed form still gives every root over the complex numbers after a short sequence of algebraic steps: divide by a to make the polynomial monic, shift the variable to remove the quadratic term, solve the resulting depressed cubic t cubed plus p t plus q equals zero, then shift the three solutions back. The sign of the depressed discriminant completely classifies the picture: three distinct real roots, a multiple real root pattern, or one real root paired with a complex conjugate pair.

How Cardano's method solves a general cubic

Begin with the general real cubic a x cubed plus b x squared plus c x plus d equals zero and insist that a is nonzero so the equation truly has degree three. Divide through by a to obtain the monic form x cubed plus A x squared plus B x plus C equals zero.

The substitution x equals t minus A over three removes the squared term and produces the depressed cubic t cubed plus p t plus q equals zero, where p and q are elementary polynomials in A, B, and C. The discriminant of that depressed cubic is Delta equals the square of q over two plus the cube of p over three. When Delta is positive, classical Cardano expressions built from two cube roots and the non-real cube roots of unity yield one real root and a complex conjugate pair.

When Delta is zero, all roots are real and at least two coincide, recovering double-root textbook cases such as a perfect cube or a squared linear factor times another linear factor. When Delta is negative—the famous casus irreducibilis—three distinct real roots appear, and the stable evaluation path uses the trigonometric identity with two times the square root of negative p over three times cosines of a third-angle arccos argument shifted by multiples of two pi over three.

After the three t values are known, every root is shifted back by subtracting A over three, restoring solutions of the original monic and therefore of the original scaled polynomial. Coefficients must be finite; NaN and Infinity are rejected. The special case a approximately zero is rejected with invalid input because Cardano would divide by zero and the problem would no longer be cubic.

Where cubic equation solvers show up in real work

Secondary-school and college algebra courses constantly assign Cardano drills, discriminant classification tables, and factor-or-formula comparisons; a single trusted oracle that returns all three roots with real and imaginary parts speeds grading and demo setup. Programming contest problems and interview warm-ups often hide a cubic under volume constraints, cubic cost models, or depressed forms after a substitution, and engineers need exact classification of three-real versus complex outcomes rather than a silent NaN from an unchecked radical.

Control theory and mechanical design solve cubic characteristic equations for stability margins and need every complex eigenvalue of a three-by-three companion matrix without leaving the closed-form path. Graphics, animation, and game engines sample cubic Bezier timing curves and solve for parameter roots of cubic polynomials along collision or silhouette queries. If you later need the quadratic formula alone, quartic Ferrari solutions, systems of two polynomial equations, or symbolic factoring over the integers, treat those as separate search intents and landings rather than overloading this cubic Cardano solver page.