So:
q_{neg}^2 + 2rq_{neg}\sqrt{\frac{F_{e,f}}{k_{C}}} - \frac{F_{e,i}r^2}{k_{C}} = 0
q_{neg}^2 + 2r\sqrt{\frac{F_{e,f}}{k_{C}}}q_{neg} = \frac{F_{e,i}r^2}{k_{C}}
x=q_{neg}, I know that, which would make:
q_{neg}^2 + 2q_{f}q_{neg} = (q_{neg} + q_{f})^2 - q_{f}^2
and that almost works, but...