The Quadratic Formula Calculator solves equations of the form ax² + bx + c = 0, giving real or complex roots with the steps shown. Fractions such as 3/4 can be used.
x = -0.5 (double root)
| Discriminant b² − 4ac | 0 |
| Root type | One repeated real root |
| Roots | x₁ = -0.5, x₂ = -0.5 |
| Vertex | (-0.5, 0) |
Calculation steps:
| Formula | x = (−b ± √(b² − 4ac)) ÷ 2a |
| Substitute | x = (−(1) ± √(1² − 4·1·0.25)) ÷ (2·1) |
| Simplify | x = (-1 ± √0) ÷ 2 |
A quadratic equation has the form ax² + bx + c = 0 with a ≠ 0. Its graph is a parabola, and its solutions (roots) are where the parabola crosses the x-axis. Quadratics describe projectile motion, areas, profit curves and much more.
x = (−b ± √(b² − 4ac)) ÷ 2a. The expression b² − 4ac is the discriminant: positive means two real roots, zero means one repeated root, negative means two complex roots.
Worked example (defaults): x² + x + 1/4 = 0. The discriminant is 1 − 1 = 0, so there is one double root x = −1/2; the equation is (x + 1/2)² = 0.
Factoring works when the equation splits into simple brackets, e.g. x² − 5x + 6 = (x − 2)(x − 3). Completing the square rewrites the equation as a(x − h)² + k = 0 and is how the formula is derived. Graphing shows the roots visually.
The vertex of y = ax² + bx + c is at x = −b/2a; it is the minimum if a > 0 and the maximum if a < 0.
What if a = 0? The equation is linear: x = −c/b.
What does i mean? i is the imaginary unit, √−1.
Can I enter decimals? Yes, or fractions like 3/4.