Question

The number of real solution of the equation $2^{\frac{x}{2}}+(\sqrt{2}+1{)}^x=(5+2\sqrt{2}{)}^{\frac{x}{2}}$ is:

• A. one
• B. two
• C. four
• D. infinite

Answer 1

The correct option is A one

$2^{x/2}+(\sqrt{2}+1{)}^x=(5+2\sqrt{2}{)}^{x/2}\Rightarrow 2^{x/2}+(3+2\sqrt{2}{)}^{x/2}=(5+2\sqrt{2}{)}^{x/2}$as($\sqrt{3+2\sqrt{2}}=1+\sqrt{2})$
Only $x=2$ satisfies this