The number of real solutions of the equation $2^{x/2}+(\sqrt{2}+1)^x=(5+2\sqrt{2})^{x/2}$, is

1

We have,

$2^{x/2}+(\sqrt{2}+1)^x=(5+2\sqrt{2})^{x/2}$

$(\sqrt{2})^x+(\sqrt{2}+1)^x=\{\sqrt{(\sqrt{2})^2+(\sqrt{2}+1)^2}\}^x$

$⇒(\sqrt{2})^x+(\sqrt{2}+1)^x=(\sqrt{5+2\sqrt{2}})^x$

$⇒(\frac{\sqrt{2}}{\sqrt{5+2\sqrt{2}}})^x+(\frac{\sqrt{2}+1}{\sqrt{5+2\sqrt{2}}})^x=1$

$⇒(\frac{\sqrt{2}}{\sqrt{5+2\sqrt{2}}})^x+(\frac{\sqrt{2}+1}{\sqrt{5+2\sqrt{2}}})^x=(\frac{\sqrt{2}}{\sqrt{5+2\sqrt{2}}})^2+(\frac{\sqrt{2}+1}{\sqrt{5+2\sqrt{2}}})^2$

$⇒x=2$

Therefore, there is only one real solution possible.