# Quartic equation solution

This calculator produces quartic equation solution using resolvent cubic.

### This page exists due to the efforts of the following people:

#### Timur

Created: 2018-02-13 14:43:18, Last updated: 2021-02-28 16:18:07 This content is licensed under Creative Commons Attribution/Share-Alike License 3.0 (Unported). That means you may freely redistribute or modify this content under the same license conditions and must attribute the original author by placing a hyperlink from your site to this work https://planetcalc.com/7715/. Also, please do not modify any references to the original work (if any) contained in this content.

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

Digits after the decimal point: 2
x1

x2

x3

x4

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 pi and qi 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

1. M. Abramovitz and I. Stegun Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables, 10th printing, Dec 1972, pp.17-18

URL copied to clipboard

#### Similar calculators

PLANETCALC, Quartic equation solution