2
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I know the answer is $2$. I guessed it. But how do you do it mathematically? $$(5-2\sqrt6)^{x/2}+ (5+2\sqrt6)^{x/2} = 10$$

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    $\begingroup$ Since $5-2\sqrt{6} = (5+2\sqrt{6})^{-1}$, your equation is equivalent to finding a root for a quadratic polynomial in $u = (5+2\sqrt{6})^{x/2}$. $\endgroup$ – achille hui Sep 12 '15 at 6:14
  • $\begingroup$ Hadn't thought of that thanks a lot. Y don't u post that as an answer $\endgroup$ – Jamal30 Sep 12 '15 at 6:18
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Let $(5+2\sqrt{6})^{x/2}=t$ then $\frac{1}{t}=(5-2\sqrt{6})^{x/2}$. Tthe equation becomes

$$t+\frac{1}{t}=10$$

$$t^2-10t+1=0$$

solve for $t$ and put $ t = (5+2\sqrt{6})^{x/2}$.

You will get 2 values of $x$ for each $t$.

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  • $\begingroup$ Just one value of $x$ for each $t$, surely? $\endgroup$ – TonyK Sep 12 '15 at 8:20
  • $\begingroup$ i didn't solve it plz edit if you feel something is wrong $\endgroup$ – Display name Sep 12 '15 at 8:22

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