Cubic Equation Solver
This cubic equation solver helps you find the roots of any third-degree polynomial of the form $$ax³ + bx² + cx + d = 0$$. Whether the equation has real or complex roots, the tool uses Cardano’s method to calculate accurate solutions instantly. It’s perfect for students, teachers, and professionals who need to solve cubic equations quickly and reliably.
Cubic Equation
Binomial Multiplication Formula
Where:
- $$t = x + \frac{b}{3a}$$
- $$p = \frac{3ac – b^2}{3a^2}$$
- $$q = \frac{2b^3 – 9abc + 27a^2d}{27a^3}$$
Then the root is: $$t = \sqrt[3]{-\frac{q}{2} + \sqrt{\left( \frac{q}{2} \right)^2 + \left( \frac{p}{3} \right)^3}} + \sqrt[3]{-\frac{q}{2} – \sqrt{\left( \frac{q}{2} \right)^2 + \left( \frac{p}{3} \right)^3}}$$
Cardano’s formula provides the real solution(s) of a depressed cubic. The full equation’s roots are then found by back-substitution.
Cubic equations are fundamental in algebra and arise in various fields such as mechanics, electronics, and 3D modeling. This calculator uses a step-by-step method based on Cardano’s formula to solve equations like $$2x³ – 4x² – 22x + 24 = 0$$. After reducing the original cubic to a simpler “depressed” form, the formula determines whether the roots are real or complex, single or multiple. The tool provides an instant and reliable way to analyze cubic behavior without manual calculations.