Question

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

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

Answer 1

Answer: (A)

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