Cubic Equation Solver
Solve any cubic equation ax³ + bx² + cx + d = 0 for its real and complex roots.
About the Cubic Equation Solver
Every cubic has three roots counting repeats: either three real roots or one real root and a complex-conjugate pair. The calculator reduces the equation to a depressed cubic, uses the trigonometric method when there are three real roots and Cardano's formula otherwise, then refines the real roots with Newton's method for full precision.
The formula
Δ = 18abcd − 4b³d + b²c² − 4ac³ − 27a²d²
- Δ > 0
- three distinct real roots
- Δ = 0
- a repeated root
- Δ < 0
- one real, two complex roots
Worked example
x³ − 6x² + 11x − 6 = 0
- Roots 1, 2 and 3, since (x − 1)(x − 2)(x − 3) expands to it.
Questions people ask
Can a cubic have no real roots?
No: it always crosses the x-axis at least once.
Why are complex roots in pairs?
With real coefficients, complex roots come in conjugate pairs a ± bi.