Example 7 Algebraic expression
$$
\sqrt{x^{2}+4}+\sqrt{(12-x)^{2}+9}
$$

The minimum value is $\qquad$
Notice:
$$
\begin{array}{l}
\sqrt{x^{2}+4}+\sqrt{(12-x)^{2}-9} \\
=\sqrt{(0-x)^{2}+[(-2)-0]^{2}}+ \\
\sqrt{(12-x)^{2}+(3-0)^{2}} .
\end{array}
$$

Establish a rectangular coordinate system as shown in Figure 8.
Let $A(0,-2)$, $B(12,3)$, and $P(x, 0)$. Then,
$$
\begin{array}{l}
|P A|=\sqrt{x^{2}+4}, \\
|P B|=\sqrt{(12-x)^{2}+9} .
\end{array}
$$

Thus, $|P A|+|P B| \geqslant|A B|=\sqrt{12^{2}+5^{2}}=13$, with equality holding if and only if point $P$ coincides with $P'$. Therefore, the minimum value of the given algebraic expression is 13.