Find shortest distance from the parabola $y=x^2-9$ to the origin. 
Find shortest distance from the parabola $y=x^2-9$ to the origin.

First, I find minima of $\sqrt{x^2+(x^2-9)^2}$, so use derivative and ...
Is have an easier way?

$$x^2+(x^2-9)^2=(x^2-9)^2+x^2-9+\frac{1}{4}+\frac{35}{4}=\left(x^2-9+\frac{1}{2}\right)^2+\frac{35}{4}\geq\frac{35}{4}.$$
The equality occurs for $x^2=\frac{17}{2},$ which says that $\frac{\sqrt{35}}{2}$ is a minimal value.
