<s>[INST] Solve $(5+2\sqrt{6})^{\frac{x}{2}} + ( 5-2\sqrt{6})^{\frac{x}{2}} = 10$ I wish to solve the equation
$$(5+2\sqrt{6})^{\frac{x}{2}} + ( 5-2\sqrt{6})^{\frac{x}{2}} = 10$$
I tried factorizing until I reached
$(\sqrt{2}+\sqrt{3})^x + (\sqrt{2}-\sqrt{3})^x = 10$ 
But from there I don't know what to do any help would be welcome 
Thanks in advance
 [/INST] Since $\sqrt{3} - \sqrt{2} > 0$, your equation should simplify to
$$ (\sqrt{3} + \sqrt{2})^x + (\sqrt{3} - \sqrt{2})^x = 10 $$
Also note that
$$ \sqrt{3} - \sqrt{2} = \frac{1}{\sqrt{3}+\sqrt{2}} $$
Let $t = (\sqrt{3}+\sqrt{2})^x$, then
$$ t + \frac{1}{t} = 10 $$
$$ t^2 - 10t + 1 = 0 $$
which gives $t = 5 \pm 2\sqrt{6}$
Therefore $x = \pm 2$
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