1. **Problem statement:** Find two possible values of $x$ such that the distance between the points $(-1, 13)$ and $(x, 9)$ is $\sqrt{65}$. 2. **Formula used:** The distance $d$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by: $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ 3. **Apply the formula:** Here, $d = \sqrt{65}$, $(x_1, y_1) = (-1, 13)$, and $(x_2, y_2) = (x, 9)$. 4. Substitute values: $$\sqrt{65} = \sqrt{(x - (-1))^2 + (9 - 13)^2} = \sqrt{(x + 1)^2 + (-4)^2} = \sqrt{(x + 1)^2 + 16}$$ 5. Square both sides to eliminate the square root: $$65 = (x + 1)^2 + 16$$ 6. Simplify: $$(x + 1)^2 = 65 - 16 = 49$$ 7. Take the square root of both sides: $$x + 1 = \pm 7$$ 8. Solve for $x$: - If $x + 1 = 7$, then $x = 6$ - If $x + 1 = -7$, then $x = -8$ **Final answer:** The two possible values of $x$ are $6$ and $-8$.