Polynomial Root Finder (degree ≤ 4)
Closed-form roots for linear, quadratic, cubic (Cardano), and quartic (Ferrari) polynomials. Complex roots supported.
Result
- Degree2
- Coefficients1.000, -3.000, 2.000
- x_12.000000
- x_21.000000
- Summary2 real roots
Step-by-step
- Discriminant Δ = b² − 4ac = 9 − 8 = 1.0000.
- Real roots: x = (−b ± √Δ) / (2a) = (3 ± 1.0000) / 2.
How to use this calculator
- Pick the degree.
- Enter coefficients from highest power down. Use 0 for missing terms (e.g. x³ − 8 = 0 → a=1, b=0, c=0, d=−8).
- Real and complex roots are both returned.
About this calculator
Closed-form (radical) root-finding for polynomials of degree 1-4. Linear and quadratic are textbook formulas. Cubic uses Cardano's method (Gerolamo Cardano, 1545) via the depressed-cubic substitution x = t − b/(3a) — when the discriminant is negative the formula technically returns three real roots through complex intermediate steps (the casus irreducibilis), so we use the equivalent trigonometric form in that case. Quartic uses Ferrari's method (Lodovico Ferrari, 1540, published by Cardano): reduce to a depressed quartic, solve a resolvent cubic to find a parameter z, then split into two quadratics. The Abel-Ruffini theorem (1824) proves there is NO general radical formula for degree ≥ 5 — those require numerical methods (Newton, Durand-Kerner, Jenkins-Traub) instead, out of scope for this tool.