Why does the solution for the voltage of this RLC circuit result in a maximum capacitor value?
Why the fear of complex numbers? Just do a distinction of cases and generate solutions for the real and the complex case. $b-a$ in the denominator will always be real, as the imaginary parts will cancel out each other, so complex numbers will only be in the exponents, wich will lead to some nice sinusoidal functions, which is expected, as this is an oscillating system. The real case just represents the overdamped case with no oscillation. If you don't know what I'm talking about have a look at this wikipedia article:
https://en.wikipedia.org/wiki/Damping_ratio