In the following manipulation
$$\frac{-(M-\frac{1}{2}\Omega)+\sqrt{(M-\frac{1}{2}\Omega)^2+2\lambda\Omega}}{\Omega} = \frac{-(M-\frac{1}{2}\Omega)^2+(M-\frac{1}{2}\Omega)^2+2\lambda\Omega}{\Omega\left[(M-\frac{1}{2}\Omega)+\sqrt{(M-\frac{1}{2}\Omega)^2+2\lambda\Omega}\right]} \; ,$$
I don't understand why the left fraction is changed to the right fraction. As I know $(a+b)^2=a^2+2ab+b^2$, but in the picture the $2ab$ term is missing. I do not understand why that is.
We want to simplify $$\frac{-(M-\frac{1}{2}\Omega)+\sqrt{(M-\frac{1}{2}\Omega)^2+2\lambda\Omega}}{\Omega} = \frac{a+b}{\Omega}$$ And the first thing to notice is that we want to get rid of the square root; in other words we want something like $b^2$ in the numerator. We can expand by $a-b$: $$ \frac{a+b}{\Omega} = \frac{(a+b)(a-b)}{\Omega (a-b)} = \frac{a^2 -b^2}{\Omega (a-b)} $$ Plugging in the original values, we obtain $$ \begin{split} \frac{-(M-\frac{1}{2}\Omega)+\sqrt{(M-\frac{1}{2}\Omega)^2+2\lambda\Omega}}{\Omega} = & \frac{(M-\frac{1}{2}\Omega)^2-\left[(M-\frac{1}{2}\Omega)^2+2\lambda\Omega \right]}{\Omega \left( -(M-\frac{1}{2}\Omega) - \sqrt{(M-\frac{1}{2}\Omega)^2+2\lambda\Omega }\right)} \\ = & \frac{2\lambda}{ (M-\frac{1}{2}\Omega) + \sqrt{(M-\frac{1}{2}\Omega)^2+2\lambda\Omega }} \\ \end{split} $$ Whether this is more useful than the first form, depends on the context.