For any equation ax² + bx + c = 0 (a ≠ 0). The discriminant b² − 4ac reveals the roots before you solve: positive → two distinct real roots, zero → one repeated real root, negative → two complex conjugate roots.
Solving quadratic equations
A quadratic equation has the form ax² + bx + c = 0. The quadratic formula, x = (−b ± √(b² − 4ac)) / 2a, solves any of them. The quantity under the root, the discriminant, determines whether the roots are real and distinct, real and repeated, or complex.
This tool also reports the parabola's vertex and the direction it opens, which is useful for graphing and optimization problems.
Worked example
Solve 2x² − 4x − 6 = 0: here a = 2, b = −4, c = −6. The discriminant is (−4)² − 4(2)(−6) = 16 + 48 = 64; √64 = 8. So x = (4 ± 8) ÷ 4, giving x = 3 and x = −1. Two distinct real roots, exactly as the positive discriminant promised.
What the discriminant tells you before solving
The quantity b² − 4ac is a preview of the answer: positive means the parabola crosses the x-axis twice (two real roots); zero means it just touches (one repeated root); negative means it never crosses, so the roots are a complex-conjugate pair — this calculator shows those too, in a ± bi form.
Vertex, axis of symmetry and graphing
The vertex sits at x = −b ÷ 2a — for the example, x = 1, and substituting back gives y = −8, so the vertex is (1, −8) and the parabola opens upward (a > 0). The axis of symmetry is the vertical line through the vertex, and the roots (when real) sit symmetrically around it — a quick way to sanity-check your answer.
When the formula beats factoring
Factoring is elegant when the roots are small integers, but the quadratic formula works on every quadratic — messy decimals, irrational roots (when the discriminant isn't a perfect square) and complex cases alike. Working with data instead of equations? The standard deviation calculator covers descriptive statistics with the same step-by-step approach.