The number of real solutions of equation `2^(x/2)+(sqrt2+1)^x=(5+2sqrt2)^(x/2)` is

Options

A. 1
B. 2
C. 4
D. infinite

Answer

A

Explanation

We have,
`2^(x//2)+(sqrt(2)+1)^(x)=(5+2sqrt(2))^(x//2)`
`(sqrt(2))^(x)+(sqrt(2)+1)^(x)={sqrt((sqrt(2))^(2)+(sqrt(2)+1)^(2))}^(x)`
`implies (sqrt(2))^(x)+(sqrt(2)+1)^(x)=(sqrt(5+2sqrt(2)))^(x)`
`implies ((sqrt(2))/(sqrt(5+2sqrt(2))))^(x)+((sqrt(2)+1)/(sqrt(5+2sqrt(2))))^(x)=1`
`implies ((sqrt(2))/(sqrt(5+2sqrt(2))))^(x)+((sqrt(2)+1)/(sqrt(5+2sqrt(2))))^(x)=((sqrt(2))/(sqrt(5+2sqrt(2))))^(2)+((sqrt(2)+1)/(sqrt(5+2sqrt(2))))^(x)`
`implies x=2`