Quadratic Formula Calculator
Solve any quadratic equation and see the discriminant, roots and vertex.
The quadratic formula
A quadratic equation has the form ax² + bx + c = 0. Its solutions are x = (−b ± √(b² − 4ac)) ÷ 2a. Enter a, b and c and the calculator solves it and shows the steps.
The discriminant tells the story
The part under the square root, D = b² − 4ac, tells you what kind of answers to expect.
- D > 0: two different real roots.
- D = 0: one repeated real root.
- D < 0: no real roots; there are two complex roots.
Example
For x² − 3x + 2 = 0, a = 1, b = −3, c = 2. D = 9 − 8 = 1, so x = (3 ± 1) ÷ 2, giving x = 2 and x = 1.
The vertex of the parabola y = ax² + bx + c is at x = −b ÷ 2a, which is also halfway between the two roots when they are real.
Frequently asked questions
What if a is zero?
Then it is not a quadratic. The calculator treats it as the linear equation bx + c = 0 and solves for x.
What are complex roots?
When D is negative, the roots involve i, the square root of −1. They come as a pair, such as −1 + 2i and −1 − 2i.
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