If $x$ is a nonnegative real number, find the minimum value of $\sqrt{x^2 +4} + \sqrt{x^2 -24x+153}$

Thinking geometrically, you want to minimise the sum of distances from $(x,0)$ to $(0,2)$ and $(x,0)$ to $(12,3)$. Reflect the point $(12,3)$ in the $x$-axis.

The total length of the two line segments from $(0,2)$ to $(x,0)$ and $(x,0)$ to $(12,-3)$ will be smallest when they form a straight line.

Hence the minimum sum of distances is the distance between $(0,2)$ and $(12,-3)$, which is $13$.

Process 1: Derivate it to zero

Process 2: Using triangle inequality,

Given,$$\sqrt{x^2 +4} + \sqrt{x^2 -24x+153}$$

$$\rightarrow\sqrt{x^2+2^2}+\sqrt{\left(-x+12\right)^2+3^2}\geq \sqrt{\left(x-x+12\right)^2+(2+3)^2} = \sqrt{{12}^2+{5}^2}$$

$$=\sqrt{169}=13$$

I.e, So,

The minimum value is $$\bbox[5px,border:2px solid red]{13}$$