The hydrogen ion concentration in weak acid of dissociation constant $K_a$ and concentration c is nearly equal to

                              $\mathrm{HA} \rightleftarrows {\mathrm{H}}_3{\mathrm{O}}^{+} + {\mathrm{A}}^{-}$

At equilibrium      $C\left(1-\alpha \right)$    $C\alpha$           $C\alpha$

$\left[{\mathrm{H}}_3{\mathrm{O}}^{+}\right] = \mathrm{C\alpha } \mathrm{and} {\mathrm{K}}_{\mathrm{a}}=\frac{\mathrm{C\alpha }\times \mathrm{C\alpha }}{\mathrm{C}(1-\mathrm{\alpha })}={\mathrm{C\alpha }}^2 \Rightarrow \mathrm{\alpha }=\sqrt{\frac{{\mathrm{K}}_{\mathrm{a}}}{\mathrm{c}}}$

$\mathrm{So} \left[{\mathrm{H}}_3{\mathrm{O}}^{+}\right] = \mathrm{C} \sqrt{\frac{{\mathrm{K}}_{\mathrm{a}}}{\mathrm{c}}} = \sqrt{{\mathrm{K}}_{\mathrm{a}}\mathrm{c}}$