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 $cos^x \,\alpha + sin^x\, \alpha = 1$
$\therefore \, x = 2$.