A cubic equation is a polynomial of degree 3, with the general form ax³ + bx² + cx + d = 0. Unlike quadratic equations, cubic equations can have 1, 2, or 3 real solutions and don't have a simple closed-form formula that most people learn in school. However, they are solvable using Cardano's formula or numerical methods.
The General Form
ax³ + bx² + cx + d = 0
Where a ≠ 0 (otherwise it's not cubic). The equation can have:
- 3 distinct real roots
- 1 real root and 2 complex conjugate roots
- A repeated root (when the discriminant equals zero)
Cardano's Formula
To use Cardano's formula, first depress the cubic (eliminate the x² term) by substituting x = t - b/(3a):
t³ + pt + q = 0
Then the roots are found using a complex formula involving the discriminant:
Δ = -4p³ - 27q²
If Δ > 0: three distinct real roots If Δ = 0: at least two equal real roots If Δ < 0: one real root and two complex conjugate roots
Worked Example
Solve x³ - 6x² + 11x - 6 = 0
By inspection or trial, we can test small integers. Testing x = 1:
1 - 6 + 11 - 6 = 0 ✓
So x = 1 is a root. Factoring out (x - 1):
(x - 1)(x² - 5x + 6) = 0
(x - 1)(x - 2)(x - 3) = 0
The three roots are x = 1, 2, 3.
Finding Roots Without Factoring
For cubic equations that don't factor nicely, use:
- Cardano's formula (algebraically exact but complicated)
- Numerical methods like Newton-Raphson (iterative, finds one root at a time)
- Graphing to estimate roots and refine with Newton-Raphson
Applications
Cubic equations appear in:
- Engineering (stress-strain analysis, fluid dynamics)
- Physics (projectile motion in resistance medium, cubic materials)
- Economics (optimization problems, production cost curves)
- Computer graphics (cubic Bézier curves)
Tips
If you suspect rational roots, use the Rational Root Theorem: any rational root p/q has p dividing d and q dividing a. This narrows your testing candidates significantly. Always verify roots by substitution.
Use our Cubic Equation Solver to find all roots instantly, whether real or complex.