A capacitor of capacitance $C$, having initial charge $\mathit{2}{Q}_{\mathit{0}}$, is connected to a battery of potential difference $V\mathit{=}\frac{{\mathit{Q}}_{\mathit{0}}}{\mathit{C}}$as shown. The work done by the battery is  

Final potential difference between the plates, $V\mathit{=}\frac{{\mathit{Q}}_{\mathit{0}}}{\mathit{C}}$
Final charge on the originally negative plate of the capacitor, $Q\mathit{\text{'}}\mathit{=}VC\mathit{=}\frac{{\mathit{Q}}_{\mathit{0}}}{\mathit{C}}C\mathit{=}{Q}_{\mathit{0}}$

Charge flowing through the battery, $\Delta Q\mathit{=}{Q}_{\mathit{0}}\mathit{-}\left(-2Q_0\right)\mathit{=}\mathit{3}{Q}_{\mathit{0}}$
Work done by battery,  $W=\Delta QV=3Q_0\frac{Q_0}{C}=\frac{3Q_0^2}{C}$