Quadratic Equation Calculator

How to use the quadratic equation calculator

1. Enter the coefficients a, b and c from the equation ax² + bx + c = 0.
2. The calculator automatically works out the discriminant and the solutions as you type.
3. Read the explanation below to understand what the discriminant tells you.

How the quadratic equation is solved

Every quadratic equation of the form ax² + bx + c = 0 can be solved with the quadratic formula:

Worked example: for x² − 3x + 2 = 0 we have a = 1, b = −3 and c = 2. The discriminant is (−3)² − 4 × 1 × 2 = 1, and since it is positive there are two real solutions: x = (3 ± 1) ÷ 2, giving x₁ = 2 and x₂ = 1. The sign of the discriminant decides everything: positive gives two solutions, zero gives one, and negative gives no real solutions at all.

The expression under the square root, b² − 4ac, is called the discriminant and determines how many real solutions the equation has:

• Positive discriminant — the equation has two real solutions.
• Discriminant equal to zero — the equation has one solution (a double root).
• Negative discriminant — the equation has no real solutions (the solutions are complex numbers).

The connection to the parabola

Graphically, y = ax² + bx + c is a parabola, and the solutions of ax² + bx + c = 0 are precisely the points where that parabola crosses the x-axis. This matches what the discriminant tells you: a positive discriminant means the parabola crosses the axis in two places, zero means it just touches the axis at a single point — the vertex — and a negative discriminant means the whole parabola lies above or below the axis without crossing it at all.

The vertex itself sits at x = −b ÷ (2a), exactly midway between the two roots when two real solutions exist. Substituting that x value back into the equation gives the y coordinate, which is the minimum value of the function when a is positive and the maximum when a is negative.

Frequently asked questions