Quadratic Formula Calculator
Every quadratic equation ax² + bx + c = 0 can be solved with one formula — no guessing, no factoring required. Enter the coefficients and get both roots (real or complex), the discriminant that predicts them, and the vertex of the parabola.
Results
Formula
How to use this calculator
- Rewrite your equation in the form ax² + bx + c = 0.
- Enter a, b and c, keeping their signs.
- Press Calculate for the roots, discriminant and vertex.
Worked example
Solve x² − 5x + 6 = 0.
- Discriminant: 25 − 24 = 1 (positive → two real roots).
- x = (5 ± 1) ÷ 2.
- x = 3 or x = 2.
Result: The roots are x = 3 and x = 2; the vertex is (2.5, −0.25).
Frequently asked questions
What if the discriminant is negative?
The equation has no real roots — the parabola never crosses the x-axis. The roots are complex numbers, which this calculator reports in a ± bi form.
How is the vertex calculated?
The x-coordinate is −b ÷ 2a (the axis of symmetry); substituting it back into the equation gives the y-coordinate.
Should I factor instead of using the formula?
Factoring is faster when it works neatly. The formula is guaranteed — it solves every quadratic, including ones with irrational or complex roots.
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