The circle has equation $x^2 + y^2 = 36$. Find the $x$-intercepts by setting $y = 0$:

$x^2 + 0^2 = 36$

$x^2 = 36$

$x = \pm 6$

The circle intersects the $x$-axis at $(-6, 0)$ and $(6, 0)$.