To solve for (x), we need to square both sides of the equation to eliminate the square root and then simplify.

[
\begin{aligned}
\cos x &= \sqrt{6x - x^2} \
(\cos x)^2 &= (\sqrt{6x - x^2})^2 \
\cos^2 x &= 6x - x^2 \
1 - \sin^2 x &= 6x - x^2 \
\sin^2 x &= x^2 - 6x + 1
\end{aligned}
]

Now we have a quadratic equation in terms of (\sin x). Let's solve for (x\:

[
\begin{aligned}
\sin^2 x &= x^2 - 6x + 1 \
\sin^2 x &= (x - 3)^2 - 8 \
\sin^2 x &= x^2 - 6x + 9 - 8 \
\sin^2 x &= x^2 - 6x + 1 \
0 &= x^2 - 6x + 1 - \sin^2 x \
\end{aligned}
]

This quadratic equation can be solved using the quadratic formula ((x = \frac{-b ± \sqrt{b^2 - 4ac}}{2a}) with (a = 1), (b = -6), and (c = 1 - \sin^2 x).