The number of roots of the equation `1+3^(x//2)=2^(x)` is

Options

A. 0
B. 1
C. 2
D. None then 2

Answer

B

Explanation

We have,
`1+3^(x//2)=2^(x)`
`implies ((sqrt(3))/(2))^(x)+((1)/(2))^(x)=1`
`implies(sqrt(3)/(2))^(x)+((1)/(2))^(x)=((sqrt(3))/(2))^(2)+((1)/(2))^(2) implies x=2`
Thus, x=2 is a solution of the given equation.
Now, let
`y=((sqrt(3))/(2))^(x)+((1)/(2))^(x)`
Clearly,`((sqrt(3))/(2))^(x)" and "((1)/(2))^(x)` are decreasing functions of x.
Therefore, `y=((sqrt(3))/(2))^(x)+((1)/(2))^(x)` is a decreasing function of x.
Cosequently, we have
`y=((sqrt(3))/(2))^(x)+((1)/(2))^(x) lt 1` for `x gt 2` and, `y gt 1` for `x lt 2`.
Hence, the given equation has just one solution.
REMARK Any equation of the form `a^(x)+b^(x)=1`, where `a^(2)+b^(2)=1` has just one solution given by x=2.