Roots of Cubic Equation Calculator

Enter coefficients, compute roots, and view clean symbolic transformations with stepwise clarity. Handle multiple cases: three real, one real, or repeated roots with notifications. See discriminant, resolvent details, and depressed substitution for insight and learning purposes. Configure precision, complex formatting, and step expansions with toggles easily. Export CSV or PDF, share, and embed results.

Input Coefficients
Equation form: a·x³ + b·x² + c·x + d = 0
Example Data
abcdNotes
1-611-6 Roots are 1, 2, 3 (three real roots).
100-1 Roots: 1 and two complex cube roots of unity.
1331 Triple root at -1 (Δ = 0, multiple root).
Results

Enter coefficients and press Compute Roots to see results.

Formula Used

Using the depressed substitution x = t - b/(3a), the cubic becomes t³ + p·t + q = 0 with

  • p = (3ac - b²)/(3a²)
  • q = (27a²d - 9abc + 2b³)/(27a³)
  • Δ = (q/2)² + (p/3)³

If Δ > 0, there is one real root and a complex conjugate pair: t = u + v, where u = ∛(-q/2 + √Δ), v = ∛(-q/2 - √Δ), and complex roots follow from multiplying by the cubic roots of unity.

If Δ = 0, at least two real roots coincide (multiple roots). If Δ < 0, there are three distinct real roots given by the trigonometric form: t_k = 2√(-p/3) · cos((φ + 2kπ)/3) with φ = arccos( (-q/2) / √(-p³/27) ). Finally, recover x = t - b/(3a).

How to Use
  1. Enter coefficients a, b, c, d for your cubic.
  2. Select decimal precision for root formatting.
  3. Click Compute Roots to calculate all roots.
  4. Review discriminant, method used, and parameters p and q.
  5. Export your results table using CSV or PDF buttons.
FAQs

It determines the nature of roots: one real plus conjugate pair, multiple real roots, or three distinct real roots.

The substitution removes the quadratic term, simplifying Cardano’s method and the trigonometric solution for three real roots.

Yes. Roots are shown as Re ± Im·i, rounded to your chosen precision.

If a = 0, it solves the resulting quadratic or linear equation automatically.

Use higher precision for sensitive comparisons or validation; lower precision for readability and quick checks.

Displayed values are formatted decimals. The method still computes exact forms internally before formatting.

When Δ < 0, Cardano’s direct radicals involve complex numbers; the trigonometric form gives three real roots cleanly.

Tip: Validate results by substituting each root into a·x³ + b·x² + c·x + d.

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