Free Maths Calculator

Quadratic Equation Calculator

Solve any quadratic equation with the formula, starting from the discriminant so you know how many solutions to expect before finding them. Whole number and fraction answers come out exactly, and surd answers show the exact form as well as the decimal. The layout follows the working expected at GCSE.

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    Worked Example

    Solve x² − 5x + 6 = 0.

    1. The coefficients:a = 1, b = −5, c = 6
    2. Discriminant:b² − 4ac = 25 − 24 = 1
    3. Positive, so two solutions:x = (5 ± √1) ÷ 2
    4. x = (5 + 1) ÷ 2 = 3 and x = (5 − 1) ÷ 2 = 2

    Answer: x = 3 or x = 2

    Try These Yourself

    Solve x² − 7x + 12 = 0.Show answerHide answer

    Answer: x = 3 or x = 4

    It factorises: (x − 3)(x − 4) = 0.

    Solve x² + 4x + 4 = 0.Show answerHide answer

    Answer: x = −2 (repeated)

    The discriminant is 16 − 16 = 0, so one repeated root.

    How many real solutions does x² + x + 5 = 0 have?Show answerHide answer

    Answer: None

    D = 1 − 20 = −19, which is negative.