Free Maths Calculator

Cubic Equation Calculator

Solve a cubic equation the way it is taught: hunt for a root with the factor theorem, divide out that factor, then solve the quadratic that remains. When no whole number or fraction root exists, the real solutions are found by sign changes instead, and a sketch of the curve shows where it crosses.

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    Worked Example

    Solve x³ − 6x² + 11x − 6 = 0.

    1. Try x = 1:f(1) = 1 − 6 + 11 − 6 = 0, so (x − 1) is a factor
    2. Divide it out:x³ − 6x² + 11x − 6 = (x − 1)(x² − 5x + 6)
    3. Factorise the quadratic:(x − 2)(x − 3) = 0
    4. So x = 1, x = 2 or x = 3

    Answer: x = 1, x = 2 or x = 3

    Try These Yourself

    Solve x³ − 7x + 6 = 0.Show answerHide answer

    Answer: x = −3, x = 1 or x = 2

    f(1) = 0, then x³ − 7x + 6 = (x − 1)(x² + x − 6).

    Show that (x + 2) is a factor of x³ + 2x² − x − 2.Show answerHide answer

    Answer: f(−2) = 0

    −8 + 8 + 2 − 2 = 0, so by the factor theorem (x + 2) is a factor.

    Solve x³ − 4x = 0.Show answerHide answer

    Answer: x = −2, x = 0 or x = 2

    Take out x: x(x² − 4) = x(x − 2)(x + 2).