TITLE: Solve $(2+\sqrt{3})^{x/2}+(2-\sqrt{3})^{x/2}=2^x$.
QUESTION [6 upvotes]: How to solve $(2+\sqrt{3})^{x/2}+(2-\sqrt{3})^{x/2}=2^x$ for $x$?

REPLY [6 votes]: 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 $x\ne2$.