Quadratic Formula Calculator

Solving ax² + bx + c = 0

This calculator solves any quadratic equation using the quadratic formula and tells you the discriminant and the parabola's vertex along the way. Enter the coefficients a, b, and c and it handles real, repeated, or complex roots automatically.

The quadratic formula

x = (−b ± √(b² − 4ac)) ÷ 2a. The part under the square root, b² − 4ac, is the discriminant, and it tells you what kind of roots to expect before you even finish.

Reading the discriminant

• Positive — two distinct real roots.
• Zero — one repeated real root.
• Negative — two complex roots (reported as a ± bi).

FAQ

What is the vertex?

The vertex is the turning point of the parabola, at x = −b ÷ 2a. It's the minimum if a is positive and the maximum if a is negative — useful for optimization problems.

What happens if I enter a = 0?

Then it isn't a quadratic — with no x² term the equation becomes linear (bx + c = 0), which has at most one root. The quadratic formula divides by a, so a must be non-zero for the tool to solve it as a quadratic.

Can it handle complex (imaginary) roots?

Yes. When the discriminant b² − 4ac is negative, there are no real roots, and the calculator reports the two complex roots in the form a ± bi instead of failing.

Does it show the steps or just the answer?

Along with the roots, it reports the discriminant and the parabola's vertex, so you can see what kind of roots to expect and where the curve turns — not just the final values.

Is it free, and does it run in my browser?

Yes to both. The solver is free with no sign-up, and all the math runs locally in your browser, so nothing you enter is sent to a server. It also works on phones and tablets.