This quadratic formula calculator solves any equation of the form ax² + bx + c = 0. It gives real or complex roots, the discriminant and the vertex of the parabola.
The quadratic formula
x = (−b ± √(b² − 4ac)) ÷ 2a
The discriminant, b² − 4ac, tells you what kind of roots to expect: positive gives two real roots, zero gives one repeated root, and negative gives two complex roots.
Worked example
For x² − 5x + 6 = 0, a = 1, b = −5 and c = 6. The discriminant is 25 − 24 = 1, so x = (5 ± 1) ÷ 2, giving x = 3 and x = 2. The vertex is at (2.5, −0.25).
Frequently asked questions
What if my equation is missing a term?
Enter 0 for that coefficient. For x² − 9 = 0, use a = 1, b = 0, c = −9.
What are complex roots?
When the discriminant is negative, the parabola never crosses the x-axis, and the roots involve i, the square root of −1. They are written as a ± bi.