I suspect that your problem is with that \(\displaystyle \sqrt{1- \frac{v^2}{c^2}}\). It is fairly easy to get to \(\displaystyle \frac{4D^2}{\Delta t^2}= c^2- v^2\) and then \(\displaystyle \frac{2D}{\Delta t}= \sqrt{c^2- v^2}\).
Now write \(\displaystyle c^2- v^2= c^2- c^2\left(\frac{v^2}{c^2}\right)\) so you can factor \(\displaystyle c^2\) out of \(\displaystyle c^2- v^2= c^2\left(1- \frac{v^2}{c^2}\right)\). Finally, take the square root: \(\displaystyle \sqrt{c^2- v^2}= \sqrt{c^2\left(1- \frac{v^2}{c^2}\right)}= c\sqrt{1- \frac{v^2}{c^2}}\).
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