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Math Help - Strange equality

  1. #1
    LoB
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    Strange equality

    x^2 / 3 + 48/x^2 = 10(x/3 - 4/x)
    after performing some actions i came to this:
    10x(x-2)(x+2) - x^4 - 144 or
    x^4 - 10x^3 + 40x +144 = 0

    any ideas what to do next, or how to solve it?
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  2. #2
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    Are you definately sure that is correct and the only information given? I can't see a way of doing that with my knowledge, but there are probably quite a few people around here better than me!
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  3. #3
    LoB
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    yeah, thats it, someone gave an idea of adding -24x^2 to both sides, so some nice grouping could be done... after some trying i got to:
    x^2 (x^2 + 1) - 10x(x-2)(x+2) + (12-x)(12+x) = 0
    but its still not enough to be able to solve :\
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  4. #4
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    Use synthetic division (trial and error), or, if you know a little bit of Calculus or derivatives, then you can use iteration by Newton's Method.

    x^2 / 3 + 48/x^2 = 10(x/3 - 4/x) -------(i)
    (x^2 / 3) +(48 / x^2) = (10x / 3) -(40 / x)
    Clear the fractions, multiply both sides by 3x^2,
    (x^2 *x^2) +(48*3) = (10x *x^2) -(40 *3x)
    x^4 +144 = 10x^3 -120x
    x^4 -10x^3 +120x +144 = 0 ----------(ii)

    So the linear or 1st degree x-term is not +40x, but it is +120x.

    You can attack (ii) by synthetic division.
    Try any integer-factor of the contant "+144"

    Or, try Newton's Method of iteration.
    x2 = x1 -[f(x1) / f'(x1)]
    x3 = x2 -[f(x2) / f'(x2)]
    and so on....
    Depending on the precision you want, even only after 3 or 4 iterations you could get one root very close to the correct root of f(x).

    In your equation above, it came to
    f(x) = x^4 -10x^3 +120x +144
    So, f'(x) = 4x^3 -30x^2 +120
    Then try a seed root of, say, x1 = 4
    etc....

    You should get x=6 as one real root in whatever way you may use.
    Check x=6 in (i),
    x^2 / 3 + 48/x^2 = 10(x/3 - 4/x) -------(i)
    6^2 / 3 +48/ 6^2 =? 10(6/3 -4/6)
    36/3 +48/36 =? 10(2 -2/3)
    12 +4/3 =? 20 -20/3
    Multiply both sides by 3,
    36 +4 =? 60 -20
    40 =? 40
    Yes, so, Ok.
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