Make x the subject of the formula: $\displaystyle v=\frac{1}{k}\sqrt{[1+(\frac{1}{x}+1)^2}/Lg$ In case I made a mistake in the coding, its v=1/k sqrt(1+(1/x +1)^2/Lg)
So we have $\displaystyle v = \frac{\sqrt{\left[1 + \left(\frac{1}{x}+1 \right)^{2} \right]}}{kLg} $
Then $\displaystyle vkLg = \sqrt{\left[1 + \left(\frac{1}{x}+1 \right)^{2} \right]} $
Or $\displaystyle v^{2}k^{2}L^{2}g^{2}-1 = \left(\frac{1}{x} +1 \right)^2 $
So $\displaystyle \left(\frac{1}{x} +1 \right)^2 = \frac{1}{x^2} + \frac{2}{x} +1 $ and $\displaystyle v^{2}k^{2}L^{2}g^{2}-2 = \frac{1}{x^2} + \frac{2}{x} $