Quadratic Equation Solver
Enter a, b and c from ax² + bx + c = 0. You get both roots, the discriminant and the vertex of the parabola.
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The formula
x = (-b ± sqrt(b² - 4ac)) / 2a
How to use the quadratic formula
Write your equation in the form ax² + bx + c = 0, identify a, b and c (including their signs), and put them into the formula. For x² - 3x + 2 = 0 we have a = 1, b = -3, c = 2.
The discriminant, b² - 4ac, comes first. Here it is 9 - 8 = 1. Then x = (3 ± 1) / 2, so x = 2 or x = 1.
What the discriminant tells you
- Positive: two different real roots. The parabola crosses the x-axis twice.
- Zero: one repeated root. The parabola just touches the axis.
- Negative: no real roots. The answers are a pair of complex numbers.
The vertex
The turning point of the parabola is at x = -b / 2a. Plug that back in to find y. For x² - 3x + 2 the vertex is (1.5, -0.25). If a is positive the vertex is the lowest point, and if a is negative it is the highest.
Common slips
Forgetting that a minus b becomes positive when b is negative is the usual one. Also remember the whole top line is divided by 2a, not just the square root.
Examples to check
Type these into the calculator above to see how it behaves. Every result below was produced by the same calculator code that runs on this page.
| What you enter | What you get |
|---|---|
| a (x² coefficient): 1; b (x coefficient): -3; c (constant): 2 | Root x₁: 2; Root x₂: 1 |
| a (x² coefficient): 2; b (x coefficient): 4; c (constant): 2 | Root x: -1 |
| a (x² coefficient): 1; b (x coefficient): 2; c (constant): 5 | Root x₁: -1 + 2i; Root x₂: -1 - 2i |
| a (x² coefficient): -1; c (constant): 9 | Root x₁: -3; Root x₂: 3 |
Quadratic Equation Solver: frequently asked questions
What if a is zero?
Then it is not a quadratic at all. It is a straight line, bx + c = 0, with the single solution x = -c / b.
Can I use decimals or fractions for a, b and c?
Decimals, yes. For fractions, type their decimal values.
How accurate is this calculator?
It uses standard double-precision arithmetic, accurate to about 15 significant digits. Results are rounded only for display.
Is what I enter in this calculator saved or sent anywhere?
No. The calculation runs in your browser, so what you enter is not sent to our servers. We only count, anonymously, whether a tool ran successfully.
Is this calculator free to use?
Yes. It is free, needs no sign-up and has no usage limit.
Last reviewed: October 3, 2026