Quartic Equation Solver

Solve ax⁴+bx³+cx²+dx+e=0 for all four roots. Real and complex roots are shown in x+yi form, with the count of real roots.

Input

Enter the coefficients of the quartic equation ax⁴+bx³+cx²+dx+e=0 to find all four roots.

a x⁴ + b x³ + c x² + d x + e = 0

a (x⁴ coefficient)

b (x³ coefficient)

c (x² coefficient)

d (x coefficient)

e (constant term)

Roots

x1 = 1

x2 = 2

x3 = 3

x4 = 4

Number of real roots

Degree of the equation

All four roots are distinct real numbers.

Details of each root

Enter the coefficients a, b, c, d and e to solve the quartic equation ax⁴+bx³+cx²+dx+e=0 for its four roots.
The solver uses an iterative numerical method (the Durand-Kerner method), so it returns complex roots in x+yi form as well as real roots.
Roots with a sufficiently small imaginary part are treated as real, and conjugate complex roots are listed together as a pair.
When the leading coefficient a is 0, the equation is solved as a cubic or lower degree polynomial instead.
Displayed values are lightly rounded for readability, so the last digits may differ slightly from the exact values.

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