Solve ax⁴+bx³+cx²+dx+e=0 for all four roots via Ferrari's resolvent-cubic method.
This solver uses Ferrari's method. The quartic is first depressed by substituting x = t − b/4, then a resolvent cubic is solved to split the depressed quartic into two quadratic factors. Each quadratic is solved with the standard formula. All four roots are reported, whether real or complex.
Depressed quartic
t⁴ + pt² + qt + r = 0 where x = t − b/(4a)
Resolvent cubic
8y³ − 4py² − 8ry + (4pr − q²) = 0
Ferrari's method reduces a quartic to a resolvent cubic. Solving the cubic yields a value used to factor the quartic into two quadratics, each solvable by the quadratic formula.
Yes — the method always yields four roots (counted with multiplicity), whether real or complex conjugate pairs.