Three, given the quadratic function $y=a x^{2}+b x+c$ whose graph has a vertex $C$, and it intersects the $x$-axis at two distinct points $A$ and $B$.
(1) If the x-coordinate of point $C$ is 3, and the distance between points $A$ and $B$ is 8, find the roots of the equation $a x^{2}-(6 a-b) x+9 a-3 b+c=0$;
(2) If the distance from point $C$ to the $x$-axis is $k$ times the distance between points $A$ and $B$, prove that $b^{2}-4 a c=15 k^{2}$.