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

Question

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

Answer 1

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 $t$.