Cubic Equation Calculator (Solve ax³ + bx² + cx + d = 0)
Enter the four coefficients to find every root.
Your numbers
Your results
Cubic Equation Calculator (Solve ax³ + bx² + cx + d = 0) by TheCalcTools
How the Cubic Equation Calculator (Solve ax³ + bx² + cx + d = 0) works
Dividing by a and substituting x = t − b/(3a) gives the depressed cubic t³ + pt + q = 0. If (q/2)² + (p/3)³ > 0 there is one real root, given by Cardano’s formula with cube roots, plus a complex pair.
If it is negative there are three real roots, found with the trigonometric form t = 2√(−p/3) cos(θ − 2πk/3). If it is zero, roots repeat.
The formula
Worked example
Using the default values shown in the calculator:
| a (x³) | 1 |
|---|---|
| b (x²) | -6 |
| c (x) | 11 |
| d (constant) | -6 |
- x³ − 6x² + 11x − 6 = 0 has roots x = 1, 2, 3.
Result:
| Roots | x = 1, 2, 3 |
|---|---|
| Discriminant Δ = −108a⁴[(q/2)² + (p/3)³] | 4.0000 |
| Root types | Three distinct real roots |
Frequently asked questions
Does every cubic have a real root?
Yes. A cubic with real coefficients always crosses the x-axis at least once.
Sources and further reading
Related calculators
More in Math · All calculators
This calculator gives estimates for information only. Check important results independently before you rely on them. See our disclaimer.