smc

The least possible value of $y$ is given at the $y$ coordinate of the vertex. The $x$- coordinate is given by $$\frac{-p}{(2)(1)}=\frac{-p}{2}$$ Plugging this into the quadratic, we get $$y=\frac{p^2}{4}-\frac{p^2}{2}+q$$ $$0=\frac{p^2}{4}-\frac{2p^2}{4}+q$$ $$0=\frac{-p^2}{4}+q$$ $$q=\frac{p^2}{4}=\boxed{\frac{p^2}{4}}$$