Quartic Equation Solver

Find all four roots of a quartic (fourth-degree) equation ax⁴ + bx³ + cx² + dx + e = 0, including repeated and complex roots. Counted with multiplicity, a quartic with real coefficients always has four roots: four real, two real plus one complex pair, or two complex pairs. The solver also writes the polynomial in factored form.

How to use this tool

1. Enter the five coefficients a to e, using 0 for missing terms.
2. Read the roots: real ones from smallest to largest, then complex ones as p + qi.
3. Check the number of real roots and any repeated roots under “Nature of roots”.
4. Use the factored form and the Vieta check in the notes to confirm the answer.

Formula

ax⁴ + bx³ + cx² + dx + e = a(x − r₁)(x − r₂)(x − r₃)(x − r₄)

Expanding the factored form gives Vieta’s relations, for example r₁ + r₂ + r₃ + r₄ = −b/a and r₁r₂r₃r₄ = e/a. Ferrari’s method solves quartics algebraically through a resolvent cubic, but the closed formula can lose accuracy in floating-point arithmetic, so this solver finds all four roots together with the Aberth–Ehrlich iteration and then pins down repeated roots using the polynomial’s derivatives.

Example

For x⁴ + 3x³ + 5x² + x − 10 = 0 the solver returns x = −2, x = 1 and the complex pair x = −1 ± 2i. The factored form is (x + 2)(x − 1)(x² + 2x + 5); the quadratic factor has discriminant 4 − 20 = −16, which is why its roots are complex. Vieta check: the roots add to −2 + 1 − 1 − 1 = −3 = −b/a, and their product is (−2)(1)(5) = −10 = e/a.