= Solution
Let $\alpha$ be the <degree of ionization>. Conservation of nuclei and <charge neutrality>, with no pre-existing <electrons> or other ions, give
$$
c_{\rm M}=(1-\alpha)c_{\rm tot},\qquad
c_{\rm M^+}=c_e=\alpha c_{\rm tot}.
$$
Substitution into the definition of $K_c$ gives
$$
\frac{\alpha^2}{1-\alpha}=y,\qquad
y=\frac{K_cc^\circ}{c_{\rm tot}}.
$$
The physical root lies between zero and one:
$$
\boxed{\alpha=\frac{\sqrt{y^2+4y}-y}{2}
=\frac{2}{1+\sqrt{1+4/y}}.}
$$
The second form avoids subtracting nearly equal terms at large $y$. The limits are $\alpha\simeq\sqrt y$ when $y\ll1$ and $\alpha\simeq1-1/y$ when $y\gg1$. The dependence on initial <number density> matters: $K_c$ alone, without $c_{\rm tot}$ and the standard-state convention, does not specify the ionized fraction.
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