<s>[INST] needs solution of the equation ${(2+{3}^{1/2}})^{x/2}$ + ${(2-{3}^{1/2}})^{x/2}$=$2^x$ $$\left(2+{3}^{1/2}\right)^{x/2} + \left(2-{3}^{1/2}\right)^{x/2} = 2^x.$$
Clearly $x = 2$ is a solution. i need others if there is any. Please help. 
 [/INST] square this equation we have
$$(2+\sqrt{3})^x+(2-\sqrt{3})^x+2=4^x$$
so
$$\left(\dfrac{2+\sqrt{3}}{4}\right)^x+\left(\dfrac{2-\sqrt{3}}{4}\right)^x+\dfrac{2}{4^x}=1$$
It is clear 
$$f(x)=\left(\dfrac{2+\sqrt{3}}{4}\right)^x+\left(\dfrac{2-\sqrt{3}}{4}\right)^x+\dfrac{2}{4^x}$$ is decreasing,because use $y=a^x,0<a<1$ is decreasing
and Note
$$f(2)=1$$
so 
$$x=2$$
or consider
$$f(x)=\left(\dfrac{\sqrt{2+\sqrt{3}}}{2}\right)^x+\left(\dfrac{\sqrt{2-\sqrt{3}}}{2}\right)^x$$
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