AMC 12 Mock Exam 2 Problem 9

Since $x^2-16x+89=(x-8)^2+5^2$, the expression is the sum of the distances from $P=(x,0)$ to $(0,3)$ and to $(8,5)$, or equally to $(0,3)$ and to the reflection $(8,-5)$. By the triangle inequality, this sum is at least the distance from $(0,3)$ to $(8,-5)$, which is $\sqrt{8^2+8^2}=8\sqrt2$. Equality holds where the segment from $(0,3)$ to $(8,-5)$ crosses the $x$-axis, at $x=3$: indeed $\sqrt{18}+\sqrt{50}=3\sqrt2+5\sqrt2=8\sqrt2$. The minimum is $\boxed{8\sqrt2}$.