Question

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

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

Answer 1

Answer: (A)

one

$2^{\frac{x}{2}}+(\sqrt{2}+1{)}^x=(5+2\sqrt{2}{)}^{\frac{x}{2}}\Rightarrow {\left(\frac{\sqrt{2}}{\sqrt{5+2\sqrt{2}}}\right)}^x+{\left(\frac{\sqrt{2}+1}{\sqrt{5+2\sqrt{2}}}\right)}^x=1$
which is of the form
$co{s}^x\alpha +si{n}^x\alpha =1$
$\therefore$ x = 2.