Solve $(2+\sqrt{3})^{x/2}+(2-\sqrt{3})^{x/2}=2^x$.

Question

How to solve $(2+\sqrt{3})^{x/2}+(2-\sqrt{3})^{x/2}=2^x$ for $x$?

Answer 1

Multiply by $(2+\sqrt 3)^{x/2}$ we get

$$(2+\sqrt 3)^x+1=2^x(2+\sqrt 3)^{x/2}=2^x((\sqrt2+\sqrt6)/2)^x=(\sqrt2+\sqrt 6)^x$$ but $(\sqrt 2+\sqrt 6)>(2+\sqrt 3)$ then if the equality is true for $x=2$ it can't be true for $x\ne2$.

Answer 2

So, by inspection, $x=2$ is a solution. It remains to show that there are no others. Since $2-\sqrt{3}<1$, for large $x$, we can essentially ignore this term, so some growth rate arguments show that one side grows faster than the other and there are no more solutions, but I'm not sure how to make this rigorous.