Can someone explain the steps to get from
$\frac{1}{[\frac{1}{2}x(1 - \sqrt{1-4/x}) -1]}$
to
$\frac{1}{2}x + \frac{1}{2}\sqrt{x(x - 4)} - 1$
when assuming x is positive?
I understand the simplification of the square root, but how did we get rid of the fraction form?
https://www.wolframalpha.com/input/?i=1%2F%5B.5x(1+-+sqrt(1-4%2Fx))+-1%5D
Hint: $\,x\sqrt{1-4/x}=\sqrt{x(x-4)}\,$ when $\,x \ge 0\,$ , then using $\,(a-b)(a+b)=a^2-b^2\,$:
$$ \begin{align} \left(\frac{x}{2} - \frac{\sqrt{x(x-4)}}{2} -1\right) \left(\frac{x}{2} + \frac{\sqrt{x(x-4)}}{2} -1\right) = \left(\frac{x}{2}-1\right)^2 - \left(\frac{\sqrt{x(x-4)}}{2}\right)^2 = \ldots \end{align} $$