First Equation $$ \Delta_{\alpha}^{*} = \left ( {b - \sqrt{\frac{(a*b)}{y(\frac{d}{c})}}} \right )_{+} $$
Derivation $$ \frac{\sqrt{c\cdot \:d}\cdot \:b-\frac{1}{y}\sqrt{a\cdot \:b}\cdot \:c}{\sqrt{c\cdot \:\:\:d}+\sqrt{a\cdot \:b}} $$
I have substituted the actual variables with variables that have been discovered for the first equation, the original equation looked like this: $$ \Delta_{\alpha}^* = \left ( {R_{\alpha } - \sqrt{\frac{k}{\gamma m_{p}}}} \right )_{+} $$
I have tried rewriting the expression myself with the variables I have found, and have not been able to replicate the derivation included in the example above. I have been trying to solve this for many hours, and I just feel as if I do not have the necessary math skills to rewrite this in the manner I am intended to.
Here is a link to the paper I have been reading from, as it may help with understanding the problem fully: https://arxiv.org/pdf/1911.03380.pdf
Relevant equations are:
PAGE 4 - 2.1 Optimal arbitrage on Uniswap
PAGE 17 - A The Uniswap arbitrage problem is convex