Formula
x = (−b ± √(b² − 4ac)) ÷ 2aD = b² − 4acVertex = (−b ÷ 2a, c − b² ÷ 4a) How it works
A quadratic equation has the form ax² + bx + c = 0 with a ≠ 0. Its graph is a parabola, and the solutions (roots) are where the parabola crosses the x-axis. The quadratic formula solves every quadratic, even ones that are hard or impossible to factor.
The discriminant D = b² − 4ac, the expression under the square root, tells you what kind of roots to expect before you finish. If D is positive there are two different real roots; if D is zero, the parabola just touches the x-axis and there is one repeated root; if D is negative the parabola never reaches the x-axis and the roots are a pair of complex numbers a ± bi.
The vertex is the turning point of the parabola, halfway between the roots at x = −b/2a. When a is positive the parabola opens upward and the vertex is the lowest point; when a is negative it opens downward and the vertex is the highest point.
Frequently asked questions
How can I check my roots?
Substitute each root back into ax² + bx + c; you should get 0. You can also check that the roots add up to −b/a and multiply to c/a.
What if a is zero?
Then there is no x² term and the equation is linear, bx + c = 0, with the single solution x = −c/b. The calculator handles that case too.
What does i mean in complex roots?
i is the imaginary unit, defined by i² = −1. Complex roots always come in conjugate pairs like 2 + 3i and 2 − 3i when a, b and c are real.