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Thread: Solve the equation

  1. #1
    Senior Member
    Nov 2007

    Solve the equation

    Solve the equaition ($\displaystyle x\in\mathbb{R}$):
    $\displaystyle \frac 1{\left(1+\sqrt{1+\sqrt{x}}\right)^4}+\frac 1{\left(1-\sqrt{1+\sqrt{x}}\right)^4}+\frac 2{\left(1+\sqrt{1+\sqrt{x}}\right)^3}+\frac 2{\left(1-\sqrt{1+\sqrt{x}}\right)^3}=0$
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  2. #2
    Super Member wingless's Avatar
    Dec 2007
    Hmm.. The solution is too long for me to write here, but let me express it. Let's call $\displaystyle \sqrt{1+\sqrt{x}} = a $

    Then the equation is,
    $\displaystyle \frac 1{\left(1+a\right)^4}+\frac 1{\left(1-a\right)^4}+\frac 2{\left(1+a\right)^3}+\frac 2{\left(1-a\right)^3}=0

    What you must do here is to equalize the denominators by expanding all the fractions as they'll all have the denominator $\displaystyle \left(1+a\right)^4$
    $\displaystyle \left(1-a\right)^4 $

    And then you can discard the common denominator, make some factorizations, simplify it as much as you can. Then this will give you a fourth degree equation which can be expressed as a quadratic equation. This will give you $\displaystyle a = \sqrt{1+\sqrt{\frac{8}{5}}}$ and you can get to $\displaystyle x = \frac{8}{5}$
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