Quartic Equation Solver
Solve ax⁴+bx³+cx²+dx+e=0 for all four roots via Ferrari's resolvent-cubic method.
Inputs
Must be non-zero
Root 1
4
Root 2
1
Root 3
3
Root 4
2
Step by step
Values used
a (x⁴ coefficient) = 1; b (x³ coefficient) = -10; c (x² coefficient) = 35; d (x coefficient) = -50; e (constant) = 24
Depressed quartic
t⁴ + pt² + qt + r = 0 where x = t − b/(4a)
Resolvent cubic
8y³ − 4py² − 8ry + (4pr − q²) = 0
Root 1
= 4
Root 2
= 1
Root 3
= 3
Root 4
= 2
How it works
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.
Formulas
Depressed quartic
t⁴ + pt² + qt + r = 0 where x = t − b/(4a)
- p
- Reduced quadratic coefficient
- q
- Reduced linear coefficient
- r
- Reduced constant
Resolvent cubic
8y³ − 4py² − 8ry + (4pr − q²) = 0
- y
- Resolvent variable
Frequently Asked Questions
What is Ferrari's method?
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.
Are all four roots always found?
Yes — the method always yields four roots (counted with multiplicity), whether real or complex conjugate pairs.