The formula of focal length for a convex lens is,

$\frac{1}{f}=\frac{1}{v}-\frac{1}{u}$, and formula for magnification, $m=\frac{v}{u}$.

For real image,

$$\begin{gathered} u_1\left(-ve\right), v_1\left(+ve\right) \\ \Rightarrow \frac{1}{f}=\frac{1}{v}-\frac{1}{u} \\ \Rightarrow \frac{1}{f}=\frac{1}{v_1}+\frac{1}{u_1} \\ \Rightarrow \frac{u_1}{f}=\frac{u_1}{v_1}+1 \\ \Rightarrow \frac{u_1}{f}=\frac{1}{m}+1 ...Eq\left(1\right) \end{gathered}$$

Similarly, for virtual image case, we can write

$$\begin{gathered} u_2\left(-ve\right), v_2\left(-ve\right) \\ \Rightarrow \frac{1}{f}=\frac{1}{v}-\frac{1}{u} \\ \Rightarrow \frac{1}{f}=-\frac{1}{v_2}+\frac{1}{u_2} \\ \Rightarrow \frac{u_2}{f}=-\frac{u_2}{v_2}+1 \\ \Rightarrow \frac{u_2}{f}=\frac{1}{-2m}+1 \\ \Rightarrow \frac{-u_2}{f}=\frac{1}{2m}-1 ...Eq\left(2\right) \end{gathered}$$

From both equations, we can write

$$\begin{gathered} \frac{u_1-u_2}{f}=\frac{1}{m}+1+\frac{1}{2m}-1 \\ \Rightarrow \frac{u_1-u_2}{f}=\frac{3}{2m} \\ \Rightarrow f=\frac{2\left(u_1-u_2\right)m}{3} \end{gathered}$$