6. A. The minimum value of the algebraic expression $\sqrt{x^{2}+4}+\sqrt{(12-x)^{2}+9}$ is
$=、 6$. A. 13.
This problem can be transformed into finding the minimum value of the sum of distances from a point $(x, 0)$ on the $x$-axis to the points $(0,2)$ and $(12,3)$ in a Cartesian coordinate system.

Since the symmetric point of $(0,2)$ with respect to the $x$-axis is $(0,-2)$, the length of the line segment connecting $(0,-2)$ and $(12,3)$ is the minimum value of the expression $\sqrt{x^{2}+4}+\sqrt{(12-x)^{2}+9}$.
In fact, when $x=\frac{24}{5}$, the expression reaches its minimum value
$$
\sqrt{(12-0)^{2}+(3+2)^{2}}=13 \text {. }
$$