How the quadratic equation solver works
Every quadratic equation ax² + bx + c = 0 with a ≠ 0 can be solved with the quadratic formula, derived by completing the square.
Discriminant D = b² − 4ac
Vertex = (−b ÷ 2a, f(−b ÷ 2a))
The discriminant tells you what kind of roots to expect before you solve. If D is positive there are two different real roots and the parabola crosses the x-axis twice. If D is zero there is one repeated root and the vertex touches the axis. If D is negative the roots are a complex conjugate pair, written p ± qi, and the parabola never meets the x-axis. The vertex is the parabola's lowest point when a is positive and its highest when a is negative.
Worked example
For x² − 3x + 2 = 0, a = 1, b = −3 and c = 2. The discriminant is 9 − 8 = 1, so there are two real roots: (3 ± 1) ÷ 2 gives x = 2 and x = 1. You can check by factoring: (x − 1)(x − 2) = 0. The vertex is at (1.5, −0.25). For x² + 2x + 5 = 0, D = 4 − 20 = −16 and the roots are −1 ± 2i.
Frequently asked questions
What is the quadratic formula?
x = (−b ± √(b² − 4ac)) ÷ 2a. It gives the solutions of ax² + bx + c = 0.
What does a negative discriminant mean?
There are no real solutions; the two roots are complex numbers and the graph does not cross the x-axis.
What if a equals zero?
Then the equation is linear (bx + c = 0) with the single solution x = −c ÷ b.
How do I find the vertex of a parabola?
The x-coordinate is −b ÷ 2a; substitute it into the equation to get the y-coordinate.
Updated October 2026. Results are estimates for education, not financial or medical advice. All math runs in your browser; nothing you type is sent anywhere.