Cubic Equation Solver
Solve ax³ + bx² + cx + d = 0 and get all three roots, real or complex.
What the Cubic Equation Solver does
A cubic always has three roots counted with multiplicity, and at least one of them is real. This solver reduces the equation to a depressed cubic and applies Cardano's method, switching to the trigonometric form when all three roots are real.
Formula
ax³ + bx² + cx + d = 0Substitute x = t − b ÷ 3a to get t³ + pt + q = 0Solve by Cardano or trigonometric form
Inputs explained
| Input | Unit | Required | Notes |
|---|---|---|---|
| a (x³) | number | Yes | — |
| b (x²) | number | Yes | — |
| c (x) | number | Yes | — |
| d (constant) | number | Yes | — |
How to use it
- Enter a (x³), b (x²), c (x) and d (constant).
- Select Calculate.
Worked example
Solve x³ − 6x² + 11x − 6 = 0.
- a
- 1
- b
- -6
- c
- 11
- d
- -6
Roots are 1, 2 and 3.
Frequently asked questions
Does a cubic always cross the x-axis?
Yes, at least once. The ends run to opposite infinities, so the curve must cross somewhere.