An ellipse has foci at $F_1 = (0,2)$ and $F_2 = (3,0).$  The ellipse intersects the $x$-axis at the origin, and one other point.  What is the other point of intersection?
The distance between the origin and $F_1$ is 2, and the distance between the origin and $F_2$ is 3, so every point $P$ on the ellipse satisfies
\[PF_1 + PF_2 = 5.\]So, if $(x,0)$ is an intercept of the ellipse, then
\[\sqrt{x^2 + 4} + \sqrt{(x - 3)^2} = 5.\]We can write this as
\[\sqrt{x^2 + 4} + |x - 3| = 5.\]If $x \le 3,$ then
\[\sqrt{x^2 + 4} + (3 - x) = 5,\]so $\sqrt{x^2 + 4} = x + 2.$  Squaring both sides, we get
\[x^2 + 4 = x^2 + 4x + 4,\]which leads to $x = 0.$  This solution corresponds to the origin.

If $x \ge 3,$ then
\[\sqrt{x^2 + 4} + (x - 3) = 5,\]so $\sqrt{x^2 + 4} = 8 - x.$  Squaring both sides, we get
\[x^2 + 4 = 64 - 16x + x^2,\]which leads to $x = \frac{15}{4}.$  Thus, the other $x$-intercept is $\boxed{\left( \frac{15}{4}, 0 \right)}.$