x² − 3x + 2 = 0
The discriminant is 9 − 8 = 1, so there are two real roots: x = 1 and x = 2. The vertex is (1.5, −0.25).
Quadratic Equation Solver
Solve ax² + bx + c = 0 with real or complex roots, the discriminant and the vertex, with working.
Result
Enter your values and select Calculate to see the result.
Runs in your browser — nothing you enter is sent to a server.
How the Quadratic Equation Solver works
Enter the coefficients a, b and c of ax² + bx + c = 0. The solver computes the discriminant b² − 4ac: if it's positive there are two real roots, if it's zero one repeated root, and if it's negative two complex roots. It also gives the vertex — the turning point of the parabola — and shows the quadratic formula with your numbers.
Formula
x = (−b ± √(b² − 4ac)) ÷ 2a
The vertex is at x = −b ÷ 2a. For very large b, the solver uses an equivalent form that avoids rounding errors.
Examples
x² + 2x + 5 = 0
The discriminant is 4 − 20 = −16, so the roots are complex: x = −1 + 2i and x = −1 − 2i.
Frequently asked questions
What if a is 0?
Then it isn't a quadratic — bx + c = 0 is linear, with x = −c ÷ b.
What does i mean?
i is the imaginary unit, √−1. Complex roots come in pairs that differ only in the sign of the imaginary part.
Can I enter decimals and negatives?
Yes. Coefficients can be any real numbers, such as −0.5 or 2.25.
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