You are given everything you need to construct the equation of the circle.
If $(a, b)$ is the center of a circle, and $r$ is its radius, then the equation of the circle is given by $$(x-a)^2 + (y-b)^2 = r^2$$
In your case, since the center is given to be $(5, 3)$, and radius is $5$, our equation of the circle is given by $$(x - 5)^2 + (y -3)^2 = (5)^2 = 25.$$
Now that we have the equation of the circle, simply show show that $(1, 0)$, and $(9,0)$ satisfy the equation. You can, alternatively, solve for $x$ when $y = 0$, as any point on the $x$-axis has a $y$-coordinate equal to $0$. Solving for $x$ when $y = 0$ will give you two solutions for $x$: $x = 1$, and $x = 9$.