About the cubic equation calculator
This free cubic equation calculator solves polynomial equations of the form ax³ + bx² + cx + d = 0. You enter the coefficients a, b, c, and d (with a ≠ 0), and the calculator finds all roots—one or three real roots, or one real and two complex—using Cardano's method. It also displays the discriminant, which determines whether you get one or three real roots. No sign-up or download is required; the tool runs in your browser and is designed to be mobile-friendly and responsive.
A cubic equation has degree 3 and always has exactly three roots in the complex numbers (counting multiplicities). Cardano's method, from the 16th century, reduces the equation to a depressed cubic and then uses a formula involving cube roots. When the discriminant is positive, there is one real root and two complex conjugate roots; when it is zero, there are three real roots with at least two equal; when it is negative, there are three distinct real roots. This is the opposite of the quadratic case, where a positive discriminant means two real roots.
Cubic equations appear in physics (e.g. certain motion and wave problems), engineering (optimization, stability), and economics (supply and demand, profit models). Solving them by hand is tedious; this calculator gives you the roots and the discriminant so you can check homework, study for exams, or explore how the coefficients affect the number and type of roots. You can try examples like x³ − 6x² + 11x − 6 = 0 (roots 1, 2, 3) or x³ − 3x + 2 = 0 (roots 1, 1, −2) to see the different cases.
Use this cubic equation calculator for homework, exam prep, or quick verification. It is free, works on phones and tablets, and shows all roots and the discriminant. For linear or quadratic equations, use our algebra or quadratic formula calculator. This page is dedicated to solving ax³ + bx² + cx + d = 0 using Cardano's method.