How to Use the Polynomial Root Finder
Enter the coefficients of a quadratic (ax² + bx + c) or cubic (ax³ + bx² + cx + d) equation and click Find Roots. The calculator returns all real and complex roots along with the discriminant for quadratics.
Quadratic Formula
D = b² − 4ac. If D > 0: two distinct real roots. D = 0: one repeated root. D < 0: two complex conjugate roots.
x = (−b ± √D) / (2a)
Cubic Equations
Every cubic has at least one real root. The solver uses Newton-Raphson across multiple starting points to find all real roots, then factors the remaining quadratic for any complex roots.
FAQ
If a = 0, the highest-degree term disappears and the equation becomes lower-degree. Enter a non-zero value for a.
The Newton-Raphson method starts from many different points to minimize misses, but very close roots might occasionally be merged. For high-precision needs, verify roots by substitution.