Quadratic equation solver
Enter a, b, and c to find the roots, the discriminant, and the vertex.
Solve ax² + bx + c = 0
The quadratic formula
For any equation ax² + bx + c = 0 with a ≠ 0, the solutions are x = (−b ± √(b² − 4ac)) ÷ 2a. The part under the square root, D = b² − 4ac, is the discriminant. It tells you what kind of answers to expect before you do any more work.
Reading the discriminant
If D is positive there are two different real roots. If D is zero there is one repeated root. If D is negative there are no real roots, and the answers are a pair of complex conjugates written a ± bi.
Vertex and graph
The vertex sits at x = −b ÷ 2a, and plugging that back in gives its height. If a is positive the parabola opens upward and the vertex is its lowest point. If a is negative it opens downward and the vertex is the highest point.
Checking your work
Substitute each root back into the original equation. You should get zero, apart from tiny rounding differences. The sum of the roots equals −b ÷ a and the product equals c ÷ a, which is a fast second check.