Solve any cubic equation ax³ + bx² + cx + d = 0 for all three roots, real or complex, with the discriminant and root type.
Result
How to use this tool
Enter a, b, c and d for ax³ + bx² + cx + d = 0.
Read all three roots, real or complex.
Use Δ to see whether the roots are distinct, repeated or complex.
Substitute a root back into the equation to confirm it gives zero.
Formula & worked example
x³ − 6x² + 11x − 6 = 0 has roots 1, 2 and 3.
Limitations — read this before you rely on it
Cubic equations only (a ≠ 0). Very large or tiny coefficients can lose precision in floating-point arithmetic.
Frequently asked questions
Cardano’s formula solves the depressed cubic t³ + pt + q = 0; this tool applies it and uses the trigonometric form when all three roots are real.
Δ > 0 means three distinct real roots, Δ = 0 means a repeated root, and Δ < 0 means one real root plus a complex conjugate pair.
Yes. A complex pair is shown as a ± bi, with the real root listed first.
Then the equation is quadratic, not cubic, so the tool asks for a non-zero a instead of giving a misleading answer.
Substitute it back: for x = 1 in the example, 1 − 6 + 11 − 6 = 0.