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Thread: If (x + 2/x)^2 = 36, then what is (x^2 + 2/x^2)^2?

  1. #1
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    If (x + 2/x)^2 = 36, then what is (x^2 + 2/x^2)^2?

    Can any kind soul help me in the below?
    Please assist in (b)

    a) Factorise t^4 - 16
    b) If (x + 2/x)^2 = 36, then what is (x^2 + 2/x^2)^2?
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  2. #2
    Super Member Bacterius's Avatar
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    Well, a) is easy, if you note that $\displaystyle 16 = 4^2$ (or $\displaystyle 2^4$).

    Second question : I guess there is another more clever way to do this but I would have solved for $\displaystyle x$ first, then evaluated.

    Since $\displaystyle (x + \frac{2}{x})^2 = 36$, we have $\displaystyle x + \frac{2}{x} = 6$, thus $\displaystyle \frac{x^2}{x} + \frac{2}{x} = 6$, therefore $\displaystyle \frac{x^2 + 2}{x} = 6$, so $\displaystyle x^2 + 2 = 6x$, and so, $\displaystyle x^2 - 6x + 2 = 0$.

    By means of the quadratic formula (for example), you find that $\displaystyle x$ equals that value and that value, then you substitute your results back into the second equation and evaluate.

    Does that make sense ?
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  3. #3
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    Dear Bacterius, thanks so much!! but i cannot get the answer of 32.
    The topic is solving quadratic equations by factorisation.
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  4. #4
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    Quote Originally Posted by ppppp77 View Post
    Can any kind soul help me in the below?
    Please assist in (b)

    a) Factorise t^4 - 16


    Remember that $\displaystyle a^2-b^2=(a-b)(a+b)$ , so in this case substitute $\displaystyle a=t^2\,,\,\,b=4$ ...


    b) If (x + 2/x)^2 = 36, then what is (x^2 + 2/x^2)^2?
    We have that $\displaystyle \left(\frac{x+2}{x}\right)^2=36\Longrightarrow \frac{x+2}{x}=\pm 6\Longrightarrow $ ....get the two possible values for x and substitute.

    Tonio
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  5. #5
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    Quote Originally Posted by ppppp77 View Post
    b) If (x + 2/x)^2 = 36, then what is (x^2 + 2/x^2)^2?
    wonder if in the second expression it's actually a $\displaystyle \frac4{x^2}$ instead of the $\displaystyle \frac2{x^2},$ because $\displaystyle \left( x^{2}+\frac{4}{x^{2}} \right)^{2}=\left( \left( x+\frac{2}{x} \right)^{2}-4 \right)^{2}=32^{2}.$
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  6. #6
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    Thanks so much Krizalid! I thought the same too.
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