jmc

We start by completing the square: $$\begin{array}{rl}9x^2+18x+7& =(3x{)}^2+18x+3^2-3^2+7\\ & =(3x+3{)}^2-9+7.\end{array}$$ Since the square of a real number is at least 0, we have $(3x+3{)}^2\ge 0$, where $(3x+3{)}^2=0$ only if $3x=-3$. Therefore, the expression is minimized when $x=\boxed{-1}.$