| Equation | Coefficients (a,b,c,d) | Expected nature | Notes |
|---|---|---|---|
| x³ − 6x² + 11x − 6 = 0 | (1, −6, 11, −6) | Three distinct real roots | Rational roots 1, 2, 3. |
| x³ + x + 1 = 0 | (1, 0, 1, 1) | One real, two complex | No rational root. Cardano works directly. |
| x³ − 3x + 2 = 0 | (1, 0, −3, 2) | Multiple root(s) | Has a repeated root (Δ = 0). |
- Cubic discriminant: Δ = 18abcd − 4b³d + b²c² − 4ac³ − 27a²d².
- Depressed cubic: x = t − b/(3a), giving t³ + p t + q = 0. p = (3ac − b²)/(3a²), q = (2b³ − 9abc + 27a²d)/(27a³).
- Cardano discriminant: Δ = (q/2)² + (p/3)³.
- Cardano root: t = ∛(−q/2 + √Δ) + ∛(−q/2 − √Δ), then x = t − b/(3a).
- Rational Root Theorem (optional): candidates are ±p/q where p | d and q | a.
- Enter a, b, c, d from ax³ + bx² + cx + d = 0.
- Select precision for the displayed roots.
- Enable rational scan if coefficients are integers.
- Press Submit to show the result above the form.
- Download CSV or PDF using the buttons.
1) What does “solvability” mean for a cubic?
Every cubic has a closed-form solution using radicals. This checker focuses on root types, rational factorability, and whether three real roots trigger casus irreducibilis.
2) What does the discriminant tell me?
For a cubic, Δ > 0 means three distinct real roots, Δ = 0 means at least one repeated root, and Δ < 0 means one real root with a complex conjugate pair.
3) Why do I sometimes see complex numbers when all roots are real?
When the depressed discriminant is negative, the cubic has three real roots, but Cardano’s formula typically passes through complex intermediate values. That situation is called casus irreducibilis.
4) What is the “rational root scan”?
It uses the Rational Root Theorem to test candidates ±p/q where p divides d and q divides a. It only runs when a, b, c, and d are integers.
5) Are the displayed roots exact?
Roots are computed numerically using Cardano’s formula and complex arithmetic, then rounded to your selected precision. For repeated roots, rounding may hide very small differences.
6) Why must a be nonzero?
If a = 0, the equation is no longer cubic. This tool is designed for third-degree polynomials; set a to a nonzero value or use a quadratic or linear solver instead.
7) Which cube-root branch is used?
The calculator uses the principal complex cube root. Different cube-root branches can permute the same three roots; the final set remains consistent up to ordering and rounding.