Rational solutions to $e^x - \frac{1}{2} = \sqrt{ x^3 + 1/4}$ Show that the only rational solution to the title curve is $x=0$. 
My attempt: Squaring both sides we have $e^{2x} - e^x + \frac{1}{4} = x^3 + \frac{1}{4}$, which yields $e^{2x} - e^x = x^3$. I suspect that the left hand side is irrational for any rational $x\neq 0$, and if so, we reach a contradiction and conclude that the only possibility is $x=0$.

You haven't sufficiently explained why the left-hand side is irrational for any rational $x \ne 0$.
Note that if $x = \frac{p}{q}$, then the equation would imply $q^3 (e^x)^2 - q^3 e^x -p^3 = 0$, in contradition to the fact that $e^x$ is transcendental for any rational $x \ne 0$.
