Quartic Equation Solver
Solve ax⁴+bx³+cx²+dx+e=0 for all four roots via Ferrari's resolvent-cubic method.
About Quartic Equation Solver
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
Worked example
With the default values below, Quartic Equation Solver returns Root 1 of 4. Change any input above and every figure here is recalculated in your browser.
Inputs used
- a (x⁴ coefficient)
- 1
- b (x³ coefficient)
- -10
- c (x² coefficient)
- 35
- d (x coefficient)
- -50
- e (constant)
- 24
Results
- Root 1
- 4
- Root 2
- 1
- Root 3
- 3
- Root 4
- 2
Step by step
- 1Values used
- 2Depressed quartic
- 3Resolvent cubic
- 4Root 1= 4
- 5Root 2= 1
- 6Root 3= 3
- 7Root 4= 2
Assumptions and limits
- The result is derived only from the 5 inputs listed above. Anything not among them is not part of the calculation.
- In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.
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.