Quartic equation solution

The calculator below solves quartic equations with a single variable. A quartic equation formula: , where a,b,c,d,e - coefficients, and x is unknown. The equation solution gives four real or complex roots. The formulas to solve a quartic equation follow the calculator.

Quartic equation

Coefficient a

Coefficient b

Coefficient c

Coefficient d

Coefficient e

Calculation precision

Digits after the decimal point: 2

Calculate

x1

 

x2

 

x3

 

x4

 

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As a first step we divide all the quartic coefficients by a to obtain the equation:

Next we solve the resolvent cubic:

We can solve it with the method described here: Cubic equation.
A single real root u₁ of this equation we'll use further for quadratic equation roots finding. If the cubic resolvent has more than one real roots, we must choose a single u₁ root in the way that gives real p and q coefficients in the formulas:
 

Then we substitute p₁, p₂,q₁,q₂, in quadratic equations in the right side of the following equation:
¹

Four roots of the two quadratic equations are the roots of original equation if the p_i and q_i signs are chosen to satisfy the following conditions:

#	Condition
1
2
3
4

Actually, we may check only the third condition, and if it is not satisfied — swap q₁ and q₂.
The solution can be verified with the calculator: Complex polynomial value calculation