Question

The number of solutions of the equation$4^x-3^{x+\frac{1}{2}}-2^{2x-1,x\in R}$ is

Answer 1

Answer: 1.5

$4^x-3^{x+\frac{1}{2}}=3^{x+\frac{1}{2}}-2^{2x-1}$
$\Rightarrow 4^x+2^{2x-1}=3^{x+\frac{1}{2}}+3^{x-\frac{1}{2}}$
$\Rightarrow 2^{2x}\left(1+\frac{1}{2}\right)=3^x\left(\sqrt{3}+\frac{1}{\sqrt{3}}\right)$
$\Rightarrow 2^{2x}\left(\frac{3}{2}\right)=3^x\left(\frac{4}{\sqrt{3}}\right)$
$2^{2x-3}=3^{x\frac{3}{2}}$
$\Rightarrow 2^{2x-3}=3^{\frac{2x-3}{2}}$
$2^{2x-3}={\left(\sqrt{3}\right)}^{2x-3}$
which holds if $2x-3=0\Rightarrow x=\frac{3}{2}$
The solution is $\left\{\frac{3}{2}\right\}$