Cubic Equation Calculator
Solve cubic equations ax³ + bx² + cx + d = 0. Uses Cardano's formula to find real roots and the discriminant.
How to Use the Cubic Equation Calculator
- Enter the coefficients a, b, c, d of ax³+bx²+cx+d=0.
- Click Solve to find the roots.
- Real roots are shown; complex roots are indicated.
Quick Reference
| From | To |
|---|---|
| x³ − 6x² + 11x − 6 = 0 | x = 1, 2, 3 |
| x³ − 1 = 0 | x = 1 |
| x³ + x = 0 | x = 0 |
| Δ > 0 | 3 distinct real roots |
| Δ < 0 | 1 real + 2 complex |
Use Cases
- •Solving cubic polynomial equations in algebra courses.
- •Finding equilibrium points in physics and engineering.
- •Analyzing cubic cost and revenue functions in economics.
Formula
Depressed cubic via substitution x = t − b/(3a). Cardano's formula for roots. Discriminant Δ = −(4p³ + 27q²).
Frequently Asked Questions
What is a cubic equation?
A polynomial equation of degree 3: ax³ + bx² + cx + d = 0, where a ≠ 0.
How many roots does a cubic have?
A cubic always has at least one real root. If the discriminant is positive, all three roots are real.
What is Cardano's formula?
A 16th-century method for solving cubic equations analytically, using cube roots.