Cubic Equation Solver
Solve cubic equations of the form ax³ + bx² + cx + d = 0 with step-by-step solutions, graphical visualization, and downloadable results
Solution
Roots Visualization
Graphical Solution
About Cubic Equations
What is a Cubic Equation?
A cubic equation is a polynomial equation of degree three. The standard form is:
ax³ + bx² + cx + d = 0
Where a, b, c, and d are coefficients, and a ≠ 0.
The Cubic Formula (Cardano’s Method)
Cubic equations can be solved using Cardano’s method, which involves reducing the cubic to a depressed cubic and then solving it using trigonometric or hyperbolic functions.
Nature of Roots
A cubic equation always has at least one real root. The discriminant (Δ) determines the nature of the roots:
- Δ > 0: Three distinct real roots
- Δ = 0: Multiple roots (either one real and one double root, or one triple root)
- Δ < 0: One real root and two complex conjugate roots
How to Use This Calculator
- Enter the coefficients a, b, c, and d in the input fields
- Click “Solve Equation” to calculate the roots
- View the step-by-step solution and root type
- Examine the visualizations of the roots and graph
- Download the solution as a PDF for your records
Example Problems
Example 1: x³ – 6x² + 11x – 6 = 0
Solution: x = 1, x = 2, x = 3 (Three distinct real roots)
Example 2: x³ – 3x² + 3x – 1 = 0
Solution: x = 1 (Triple root)
Example 3: x³ + x + 1 = 0
Solution: One real root and two complex roots