Note that this equation is of the form $a^2 + ab = 0$, which factors to $a(a + b) = 0$. Plugging in $a = x - \frac{3}{4}$ and $b = x - \frac{1}{2}$ gives:
$(x - \frac{3}{4})(x - \frac{3}{4} + x - \frac{1}{2}) = 0$
$(x - \frac{3}{4})(2x - \frac{5}{4}) = 0$
The roots are $x = \frac{3}{4}$ nad $x = \frac{1}{2} \cdot \frac{5}{4}$. The smaller root is $\frac{5}{8}$, which is option $\boxed{\textbf{(C)}}$.