Enter Coefficients for a·x³ + b·x² + c·x + d = 0
Results
Enter coefficients and press Calculate Roots to see results.
| Item | Value |
|---|---|
| No results available. | |
Formula Used
We solve a·x³ + b·x² + c·x + d = 0 with Cardano's method.
- Compute Δ₀ = b² − 3ac and Δ₁ = 2b³ − 9abc + 27a²d.
- Form Δ = Δ₁² − 4Δ₀³ and C = cube_root((Δ₁ + √Δ)/2).
- Let ω = −1/2 + i√3/2, a primitive cube root of unity.
- Roots: xk = −(b + ωᵏC + Δ₀/(ωᵏC)) / (3a) for k = 0,1,2.
How to Use This Calculator
- Enter coefficients a, b, c, d for your cubic equation.
- Click Calculate Roots to compute exact complex roots.
- Review discriminant and the nature of roots for interpretation.
- Use Download CSV or Download PDF to save results.
- Try an example by clicking a Load button below.
Example Data
| Name | a | b | c | d | Notes | Action |
|---|---|---|---|---|---|---|
| (x − 1)(x − 2)(x − 3) | 1 | -6 | 11 | -6 | Three distinct real roots: 1, 2, 3 | |
| (x + 1)³ | 1 | 3 | 3 | 1 | Triple root at -1 | |
| x³ − 3x + 1 | 1 | 0 | -3 | 1 | Three real roots (casus irreducibilis) | |
| x³ + x² + x + 1 | 1 | 1 | 1 | 1 | One real root, complex conjugate pair | |
| 2x³ − 4x + 2 | 2 | 0 | -4 | 2 | Multiple root scenario |
Root Classification by Discriminant
Use Δ to interpret the qualitative behavior of solutions.
| Discriminant Δ | Nature of Roots | Comments |
|---|---|---|
| Δ > 0 | Three distinct real roots | All roots real and unequal |
| Δ = 0 | Multiple roots | At least two roots coincide |
| Δ < 0 | One real, two complex conjugates | Pair has equal real parts |
Vieta’s Relations Check
These identities link coefficients to roots. We compute both sides.
| Relation | From Coefficients | From Computed Roots |
|---|---|---|
| Enter coefficients and calculate to see Vieta checks. | ||
Frequently Asked Questions
It solves a·x³ + b·x² + c·x + d = 0 with nonzero a.
Yes. Results include exact complex values when the discriminant is negative.
It determines the nature of roots: three real, repeated, or one real with complex conjugates.
Values are computed using stable complex arithmetic and formatted to high precision.
Degenerate cases with a=0 reduce to quadratic or linear equations automatically.
Use the CSV or PDF buttons to download a copy of the results table.