Cubic Equation Solver

Cardano's formula is the algebraic solution for cubic equations $ax^3 + bx^2 + cx + d = 0$, published in 1545. It first reduces the cubic to a depressed form $t^3 + pt + q = 0$ by substituting $x = t - \frac{b}{3a}$, then solves using cube roots.

A cubic equation always has exactly three roots (counting multiplicity). It has either three real roots or one real root and two complex conjugate roots. The cubic discriminant determines which case applies.

The casus irreducibilis occurs when all three roots of a cubic are real, yet Cardano's formula requires computing with complex cube roots. This is not a defect but a proven mathematical necessity. The calculator handles this case using trigonometric substitution.

A depressed cubic has the form $t^3 + pt + q = 0$ with no squared term. Any general cubic $ax^3 + bx^2 + cx + d = 0$ can be converted to depressed form by the substitution $x = t - \frac{b}{3a}$, which eliminates the $x^2$ term.

The cubic discriminant is $\frac{q^2}{4} + \frac{p^3}{27}$. When it is positive, the cubic has one real root and two complex conjugate roots. When it equals zero, all roots are real with at least two equal. When it is negative, all three roots are distinct and real.