Question

For what value of $p$ does the equation $p{2}^x+2^{-x}=5$ possess a unique solution.

Answer 1

$p{2}^x+2^{-x}=5$

$\Rightarrow p{2}^x+\frac{1}{2^x}=5$

$\Rightarrow p{2}^{2x}-5\left(2^x\right)+1=0$(quadratic in $2^x$)

For unique solution, $D=0$

$\Rightarrow 25-4p=0$

$\Rightarrow 4p=25$

$\Rightarrow p=\frac{25}{4}$