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Basic algebra question

  1. Jun 24, 2010 #1
    1. The problem statement, all variables and given/known data

    Given that a-b = 5 and ab = 2, what is the value of a^4 + b^4?

    2. Relevant equations

    3. The attempt at a solution

    Doing the math I know that
    [tex]a_{1} = \frac{5+\sqrt{33}}{2}[/tex]
    [tex]a_{2} = \frac{5-\sqrt{33}}{2}[/tex]

    [tex]b_{1} = \frac{-5+\sqrt{33}}{2}[/tex]
    [tex]b_{2} = \frac{-5-\sqrt{33}}{2}[/tex]

    So my question is: there is any fast way to do [tex]a^{4} + b^{4}[/tex]?
  2. jcsd
  3. Jun 24, 2010 #2


    User Avatar
    Science Advisor

    You can expand a^4+b^4 so you never have to solve for a and b, as follows:
    [tex]a^4+b^4=(a-b)^4+4a^3b-6a^2b^2+4ab^3=(a-b)^4+4ab(a^2+b^2)-6a^2b^2=(a-b)^4-6(ab)^2+4ab((a-b)^2+2ab) = (a-b)^4+2(ab)^2+4ab(a-b)^2[/tex]

    Now you just plug in what you know for (a-b) and ab.
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