Polynomial Roots Calculator
Complex roots named, and no cancellation error.
Work out Polynomial Roots. Complex roots named, and no cancellation error. Refuses out-of-range input instead of guessing.
Roots of the quadratic
3, 2
All roots are real
The quadratic roots are computed in the numerically stable form rather than as (−b ± √D)/2a. When b² dominates 4ac the two expressions are mathematically identical and the textbook one loses most of its significant figures to cancellation — a real failure that shows up on ordinary inputs, not an exotic one. The discriminant tells you the shape of the answer before you compute it: positive means two distinct real roots, zero means one repeated, negative means a conjugate pair. Complex roots are named rather than dismissed as "no solution". A quadratic with a negative discriminant has two perfectly good roots, and refusing to state them describes a limitation of the calculator rather than a property of the maths.
How the Polynomial Roots Calculator works
Roots of a quadratic or cubic, exactly, including complex ones. The quadratic uses the numerically stable form rather than the textbook formula, which loses most of its significant figures on perfectly ordinary inputs.
Also known as: solve a quadratic equation · roots of a cubic · quadratic formula calculator · what if the discriminant is negative
Frequently asked questions
What does the discriminant tell me?
The shape of the answer before you compute it. Positive means two distinct real roots, zero means one repeated root, negative means a conjugate pair of complex roots.
Why not use the standard quadratic formula?
Because when b² dominates 4ac, computing (−b + √D)/2a subtracts two nearly equal numbers and destroys precision. The stable form computes one root that way and the other from the product of roots, which is exact.
What if the roots are complex?
They are shown. A quadratic with a negative discriminant has two perfectly good roots, and reporting "no solution" describes a limitation of the calculator rather than a property of the maths.
What is the vertex of a parabola?
Its turning point, at x = −b/2a. It sits exactly halfway between the two roots when they are real, which is a useful check.
How are cubics solved here?
By reducing to a depressed cubic and applying either Cardano's formula or the trigonometric method, depending on the discriminant. Both are exact rather than iterative.
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