# Solving $(5-2\sqrt6)^{x/2}+ (5+2\sqrt6)^{x/2} = 10$

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

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

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 $t$.
• Just one value of $x$ for each $t$, surely? – TonyK Sep 12 '15 at 8:20