Quadratic Equation Calculator
Roots, discriminant and vertex — computed stably.
Work out Quadratic Equation. Roots, discriminant and vertex — computed stably. Says what the result does not mean.
Roots
2 and 3
Two distinct real roots
The discriminant decides everything: positive gives two real roots, zero gives one repeated root at the vertex, negative gives a conjugate pair of complex ones. This uses the numerically stable form rather than the textbook formula. When b² is much larger than 4ac, the direct version subtracts two nearly equal numbers for one root and loses most of its significant digits — for x² + 10⁸x + 1 it returns 0 instead of −10⁻⁸. Computing the well-conditioned root first and getting the other from c/a avoids that entirely. The vertex sits at x = −b/2a, halfway between the roots when they are real.
How the Quadratic Equation Calculator works
Enter the three coefficients for both roots, the discriminant and the vertex. It uses the numerically stable form of the formula rather than the textbook one, which loses most of its precision on one root whenever b² is much larger than 4ac.
Also known as: quadratic formula calculator · discriminant calculator · solve ax2 + bx + c · roots of a quadratic calculator
Frequently asked questions
What is the quadratic formula?
x = (−b ± √(b² − 4ac)) / 2a. The part under the root is the discriminant, and its sign determines everything about the roots.
What does the discriminant tell me?
Positive gives two distinct real roots, zero gives one repeated root at the vertex, negative gives a conjugate pair of complex roots and a parabola that never touches the x-axis.
Why does this sometimes differ from my calculator?
The textbook formula subtracts two nearly equal numbers for one root when b² greatly exceeds 4ac, and catastrophic cancellation destroys the digits. For x² + 10⁸x + 1 the naive route returns 0 instead of −10⁻⁸. This page computes the well-conditioned root first and derives the other from c/a.
What happens if a is zero?
There is no x² term, so it is a linear equation rather than a quadratic and has the single root −c/b. This page refuses it rather than dividing by zero.
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