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Quartic Equation Solver

Solve ax⁴ + bx³ + cx² + dx + e = 0 for all four roots — real or complex — using Ferrari's method.

Nature
Four real roots
Root 1
4
Root 2
1
Root 3
3
Root 4
2

How it works

This is the quartic sibling of this site's Cubic Equation Solver, one step further up the same ladder as the Quadratic Equation Solver. Dividing through by a and substituting x = y − B/4 (B = b/a) eliminates the x³ term, leaving a depressed quartic y⁴ + py² + qy + r = 0.

When the linear term vanishes (q = 0), the depressed quartic is a 'biquadratic' — substituting w = y² turns it into an ordinary quadratic in w, and each of its two roots gives y = ±√w directly.

Otherwise, Ferrari's method introduces an auxiliary value m chosen so the quartic splits into two ordinary quadratics. That m has to satisfy a cubic equation of its own (the 'resolvent cubic'), solved the same way this site's Cubic Equation Solver solves any real cubic — which always has at least one real root, so a valid m always exists. Once m is known, the two resulting quadratics are solved with the standard quadratic formula (using complex arithmetic, since the intermediate values can be complex even when the final roots turn out real), and shifting back by x = y − B/4 gives the quartic's four roots.

FAQ

Why do quartics sometimes have 0, 2, or 4 real roots, unlike cubics which always have at least one?

A quartic's graph goes to +∞ on both ends when a > 0 (or −∞ on both ends when a < 0), so unlike a cubic's graph — which is forced to cross zero by going from −∞ to +∞ — a quartic's graph can stay entirely on one side of the x-axis, dip down and come back up without crossing, or cross up to four times. That's why this calculator can report four real roots, two real roots plus one complex conjugate pair, or two complex conjugate pairs, but never exactly one or three real roots (complex roots of a real polynomial always come in conjugate pairs, so the real-root count is always even).

What does the 'biquadratic' shortcut mean, and when does it apply?

It applies whenever the depressed quartic's y term vanishes (q = 0) — which happens, for example, whenever the original equation has no x³ or x term relative to its own symmetry, like x⁴ − 5x² + 4 = 0. In that case the equation only involves x² and x⁴, so treating x² as a single unknown turns it into a quadratic, which is simpler and more numerically direct than the general resolvent-cubic route.

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