Quartic equation solution
This calculator produces quartic equation solution using resolvent cubic.
The calculator below solves quartic equation with 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.
Save the calculation to reuse next time, to extension embed in your website or share share with friends.
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 u_{1} 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 u_{1} root in the way that gives real p and q coefficients in the formulas:
Then we substitute p_{1}, p_{2},q_{1},q_{2}, 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 third condition, and if it is not satisfied — just swap q_{1} and q_{2}.
The solution can be verified with the calculator: Complex polynomial value calculation

M. Abramovitz and I. Stegun Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables, 10th printing, Dec 1972, pp.1718 ↩
Comments