Calculadora de Equação Cúbica (raiz real de x³ + px + q = 0)
- Created by
- Renato Passos, Eng. de Software
- Reviewed by
- Renato Passos, Eng. de Software
Last updated: Apr 18, 2026
Formula
x = ∛(−q/2 + √D) + ∛(−q/2 − √D) com D = (q/2)² + (p/3)³
About this calculator
The Cubic Equation Calculator solves the real root of the depressed form x³ + p·x + q = 0 using Cardano's formula. This formula applies when the discriminant D = (q/2)² + (p/3)³ is greater than or equal to zero, ensuring a single real root. The root is computed as the sum of two cube roots: ∛(−q/2 + √D) + ∛(−q/2 − √D). The calculator simplifies the process, avoiding manual errors.
How it works: the user inputs the coefficients p and q of the depressed cubic equation. The tool calculates the discriminant D and, if D ≥ 0, applies Cardano's formula to obtain the real root. If D < 0, the result is displayed as complex, because in this case the equation has three real roots that require trigonometric calculations. The focus here is only on the real root for D ≥ 0.
When to use? In engineering, physics, and mathematics problems involving cubic equations, such as vibration analysis, electrical circuits, and economic models. The depressed form appears after eliminating the quadratic term by substitution. It is useful for quickly finding the real solution without solving complex systems.
Cautions: the calculator only works for equations in the form x³ + px + q = 0. Make sure the equation is in this form before using. If the discriminant is negative, the real root is not unique; in this case, trigonometric methods are needed. The calculator does not handle complex coefficients.
Frequently asked questions
What is a depressed cubic equation?
It is a cubic equation of the form x³ + px + q = 0, obtained after eliminating the quadratic term through a variable substitution.
How do I obtain the depressed form from a general cubic?
For a general cubic ax³ + bx² + cx + d = 0, substitute x = y - b/(3a). This removes the quadratic term and results in y³ + py + q = 0.
What does a negative discriminant mean?
If D < 0, the equation has three real roots, but Cardano's formula involves complex numbers. In this case, use trigonometric methods to find the real roots.
Can I use this calculator for equations with complex coefficients?
No, the calculator is designed only for real coefficients p and q. For complex coefficients, other tools are needed.
Does the calculator provide all roots?
No, it provides only the real root when D ≥ 0. To obtain all roots, including complex ones, more complete methods are required.