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Math Help - Find roots of function

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    Find roots of function

    Firstly this question relates to an assignment so please do not provide answer.

    I am asked to find the roots of a function which appears to be a cubic function.

    Function

    \frac{x^3}{3} - x^2 - 3x

    I am given a graph which clearly shows 3 roots, which are approx -2, 0 and 5. However I need to work out the exact points.

    I have read the The "Cubic Formula" which talks about the cubic formula however I honestly cannot get my head around it. Does anyone know of a better link which explains it better or can explain it better ?

    Unfortunately the book I have been given doesn't touch on this formula.
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    Quote Originally Posted by gk99 View Post
    Firstly this question relates to an assignment so please do not provide answer.

    I am asked to find the roots of a function which appears to be a cubic function.

    Function

    \frac{x^3}{3} - x^2 - 3x

    I am given a graph which clearly shows 3 roots, which are approx -2, 0 and 5. However I need to work out the exact points.

    I have read the The "Cubic Formula" which talks about the cubic formula however I honestly cannot get my head around it. Does anyone know of a better link which explains it better or can explain it better ?

    Unfortunately the book I have been given doesn't touch on this formula.
    no need for the cubic formula ...

    \frac{x^3}{3} - x^2 - 3x = 0

    factor out an x from each term ...

    x \left(\frac{x^2}{3} - x - 3\right) = 0

    now you know one root is x = 0 ... solve the quadratic factor for the other two roots.
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  3. #3
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    Quote Originally Posted by gk99 View Post
    Firstly this question relates to an assignment so please do not provide answer.

    I am asked to find the roots of a function which appears to be a cubic function.

    Function

    \frac{x^3}{3} - x^2 - 3x

    I am given a graph which clearly shows 3 roots, which are approx -2, 0 and 5. However I need to work out the exact points.

    I have read the The "Cubic Formula" which talks about the cubic formula however I honestly cannot get my head around it. Does anyone know of a better link which explains it better or can explain it better ?

    Unfortunately the book I have been given doesn't touch on this formula.
    If you are finding the roots, you are finding the values of x that make the equation

    \frac{x^3}{3} - x^2 - 3x = 0 true.


    Multiply both sides by 3 to get rid of the fraction...

    x^3 - 3x^2 - 9x = 0


    Now factorise and use the Null Factor Law to solve for x.


    x^3 - 3x^2 - 9x = 0

    x(x^2 - 3x - 9) = 0


    So x = 0 or x^2 - 3x - 9 = 0.


    The first root is obvious, the other 2 require either completing the square or using the Quadratic Formula...

    x^2 - 3x - 9 = 0

    x^2 - 3x + \left(-\frac{3}{2}\right)^2 - \left(-\frac{3}{2}\right)^2 - 9 = 0

    \left(x - \frac{3}{2}\right)^2 - \frac{9}{4} - \frac{36}{4} = 0

    \left(x - \frac{3}{2}\right)^2 - \frac{45}{4} = 0

    \left(x - \frac{3}{2}\right)^2 = \frac{45}{4}

    x - \frac{3}{2} = \pm \sqrt{\frac{45}{4}}

    x - \frac{3}{2} = \pm \frac{3\sqrt{5}}{2}

    x = \frac{3}{2} \pm \frac{3\sqrt{5}}{2}.


    So the three roots are

    x = \frac{3 - 3\sqrt{5}}{2}, x = 0, x = \frac{3 + 3\sqrt{5}}{2}.
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