Cubic Equation Calculator
Solve any cubic equation of the form ax³ + bx² + cx + d = 0. This calculator finds all three roots, whether they are real or complex, computes the discriminant, and identifies the root type.
A cubic equation is a third-degree polynomial equation. Unlike quadratics, cubics always have at least one real root. The general solution uses Cardano's method, published in 1545.
The Discriminant:
The discriminant D = -4p³ - 27q² (where p and q come from the depressed cubic) determines the root types:
- D > 0: Three distinct real roots
- D = 0: A repeated root (one double root plus one single root, or a triple root)
- D < 0: One real root and two complex conjugate roots
Depressed Cubic:
The substitution x = t - b/(3a) eliminates the x² term, converting ax³ + bx² + cx + d = 0 into the simpler form t³ + pt + q = 0. This depressed cubic is solved using Cardano's formula or, when there are three real roots, the trigonometric method.
Applications:
Cubic equations appear in engineering (beam deflection), physics (equations of state for gases), economics (cost optimization), and computer graphics (Bezier curves). They are also the simplest polynomials where complex roots first become unavoidable in the solution process.