About the quadratic formula calculator
A quadratic equation has the form ax² + bx + c = 0 with a not equal to 0. Enter the three coefficients and the calculator applies the quadratic formula to find the values of x that make the equation true.
The discriminant b² − 4ac decides the kind of answer. If it is positive there are two different real roots. If it is zero there is one repeated root. If it is negative there are no real roots, and the roots are a pair of complex numbers shown as p ± qi.
You also get the vertex, the axis of symmetry, the direction in which the parabola opens, the y-intercept and the sum and product of the roots. If a is 0 the equation is linear and the calculator says so.
Formula
- x = (−b ± √(b² − 4ac)) ÷ 2a
- Discriminant D = b² − 4ac
- Vertex: h = −b ÷ 2a, k = c − b² ÷ 4a
- Sum of roots = −b ÷ a; product of roots = c ÷ a
Example
x² − 3x + 2 = 0 has D = 9 − 8 = 1, so x = (3 ± 1) ÷ 2, giving 2 and 1. For x² + 2x + 5 = 0, D = −16, so x = −1 ± 2i.