Quadratic Formula Calculator

Solve any quadratic equation of the form ax² + bx + c = 0. Enter a, b and c to get the roots, the discriminant and the vertex, with the working shown.

x² − 3x + 2 = 0

Solutions
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Discriminant–
Vertex–
Opens–

    The quadratic formula

    x = (−b ± √(b² − 4ac)) ÷ 2a

    A quadratic equation has the form ax² + bx + c = 0, where a is not zero. The formula gives the values of x that make the equation true, which are also the points where the parabola y = ax² + bx + c crosses the x-axis.

    Step by step: x² − 3x + 2 = 0

    1. Identify a = 1, b = −3, c = 2.
    2. Discriminant: (−3)² − 4 × 1 × 2 = 9 − 8 = 1.
    3. Square root of the discriminant: √1 = 1.
    4. x = (3 ± 1) ÷ 2, so x = 2 or x = 1.

    Check: (2)² − 3(2) + 2 = 0 ✓ and (1)² − 3(1) + 2 = 0 ✓.

    What the discriminant tells you

    b² − 4acSolutionsGraph
    PositiveTwo different real rootsCrosses the x-axis twice
    ZeroOne repeated real rootTouches the x-axis at the vertex
    NegativeTwo complex rootsNever touches the x-axis

    Other ways to solve quadratics

    • Factoring: fastest when it works. x² − 3x + 2 = (x − 1)(x − 2).
    • Completing the square: rewrites the equation as (x − h)² = k, which also reveals the vertex.
    • Graphing: the x-intercepts of the parabola are the real solutions.

    The quadratic formula always works, which makes it the safest choice when an equation does not factor neatly.

    Last reviewed October 2026.

    Frequently asked questions

    What is the quadratic formula?

    For an equation ax² + bx + c = 0, the solutions are x = (−b ± √(b² − 4ac)) ÷ 2a. It works for every quadratic equation, even ones that cannot be factored easily.

    What does the discriminant tell me?

    The discriminant is b² − 4ac. If it is positive there are two real solutions, if it is zero there is one repeated solution, and if it is negative there are two complex solutions and the graph never crosses the x-axis.

    What are complex roots?

    When the discriminant is negative, the square root of a negative number appears. The solutions are written with i, where i² = −1, for example −1 ± 2i. The calculator shows these automatically.

    Why can't a be zero?

    If a is 0, the x² term disappears and the equation is linear, bx + c = 0, with the single solution x = −c ÷ b. The quadratic formula would divide by zero.

    What is the vertex of a parabola?

    The vertex is the turning point of the graph y = ax² + bx + c. Its x-coordinate is −b ÷ 2a. If a is positive the vertex is the lowest point; if a is negative it is the highest point.