$U _{ f }=\left(\rho Sx _{ f }\right) g \cdot \frac{ x _{ f }}{2} \times 2$

By volume conservation

$Sx _{1}+ Sx _{2}= S \left(2 x _{ f }\right)$

$x_{f}=\frac{x_{1}+x_{2}}{2}$

$\Delta U =\rho \operatorname{Sg}\left[\left(\frac{ x _{1}^{2}}{2}+\frac{ x _{2}^{2}}{2}\right)- x _{ f }^{2}\right]$

$=\rho \operatorname{Sg}\left[\frac{ x _{1}^{2}}{2}+\frac{ x _{2}^{2}}{2}-\left(\frac{ x _{1}+ x _{2}}{2}\right)^{2}\right]$

$=\frac{\rho Sg }{2}\left[\frac{ x _{1}^{2}}{2}+\frac{ x _{2}^{2}}{2}- x _{1} x _{2}\right]$

$=\frac{\rho Sg }{4}\left( x _{1}- x _{2}\right)^{2}$