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Quadratic Inequalities Calculator

Solve a quadratic inequality with the three step method examiners look for: find where the matching equation equals zero, sketch the U shaped graph through those roots, then read off where the curve sits below or above the x axis. The shaded sketch shows the answer, given in inequality and set notation.

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Method, Step by Step

    Worked Example

    Solve x² − x − 6 < 0.

    1. Solve x² − x − 6 = 0:(x − 3)(x + 2) = 0, so x = 3 or x = −2
    2. Sketch the U shape crossing the x axis at −2 and 3
    3. Less than 0 means below the x axis, which is between the roots: −2 < x < 3

    Answer: −2 < x < 3

    Try These Yourself

    Solve x² − 9 ≥ 0.Show answerHide answer

    Answer: x ≤ −3 or x ≥ 3

    Roots at ±3, and greater than 0 means the two outside pieces.

    Solve x² + 2x − 8 ≤ 0.Show answerHide answer

    Answer: −4 ≤ x ≤ 2

    (x + 4)(x − 2) = 0 gives the roots, then take the part between them.

    List the integers that satisfy x² < 10.Show answerHide answer

    Answer: −3, −2, −1, 0, 1, 2, 3

    √10 is about 3.16, so −3.16 < x < 3.16.