Quartic equation solution
This calculator produces quartic equation solution using resolvent cubic.
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.
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:
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:
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
M. Abramovitz and I. Stegun Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables, 10th printing, Dec 1972, pp.17-18 ↩