How I can solve this : $-2^x+3^{x/2}+1=0$ without using numerical ways? This equation :  $-2^x+3^{x/2}+1=0$ is confusing me , However it has only one solution which it is an integer $ x=2$ , But i can't resolve it using clear way , I have used the varibale change $y= x/2$ in order to transforme it in equation of degree 2 but am regret that is not such polynomial for which solving it using standard Method , Any way without using numerical methods, I feel the only way work is to use Lambert Function  ?

Using the equivalent equation as of the hint of zeraoulia rafik, one observes that the equation
$$
-1+\left(\frac{\sqrt3}2\right)^x+\left(\frac12\right)^x=0
$$
is constant or strictly monotonously falling in all terms on the left side, so that also the whole left side as their sum is strictly falling.. Which proves that there can only be one root, which was already found by inspection to be $x=2$.
