how to derive equilibrium concentrations from initial conditions for mass action reactions?
Here is the expression from the question:
$\displaystyle\frac{(A_0 - [C]_{eq})(B_0 - [C]_{eq})}{[C]_{eq}} = K_{eq}$
Get rid of the fraction:
$(A_0 - [C]_{eq})(B_0 - [C]_{eq}) = K_{eq} {[C]_{eq}}$
Distribute the sums in the product:
$A_0 B_0 - [C]_{eq}(A_0 +  B_0)  + [C]_{eq}^2 = K_{eq} {[C]_{eq}}$
Sort the terms and you have your quadratic equation:
$A_0 B_0 - [C]_{eq}(A_0 +  B_0 - K_{eq})  + [C]_{eq}^2 =  0$