<s>[INST] How do I simplify $\frac{\sqrt{21}-5}{2} + \frac{2}{\sqrt{21} - 5}$? 
How do I simplify the following equation?
$$\frac{\sqrt{21}-5}{2} + \frac{2}{\sqrt{21} - 5}$$

I have no idea where to start. If I multiply either fraction by its denominator I will still end up with a square root. I know the end result should be $-5$.
 [/INST] $$\begin{align} \frac{\sqrt{21} - 5}{2} + \frac{2}{\sqrt{21} - 5} &= \frac{\sqrt{21} - 5}{2} + \frac{1}{\frac{\sqrt{21} - 5}{2}} \\ &= \frac{\big(\frac{\sqrt{21} - 5}{2}\big)^2 + 1}{\frac{\sqrt{21} - 5}{2}} \\ &= \frac{2\big(\frac{(\sqrt{21} - 5)^2}{4} + 1\big)}{\sqrt{21} - 5} \\ &= \frac{\frac{21 + 25 - 10\sqrt{21}}{2} + 2}{\sqrt{21} - 5} \\ &= \frac{\frac{46 + 2(2 - 5\sqrt{21})}{2}}{\sqrt{21} - 5} \\ &= \frac{23 + 2 - 5\sqrt{21}}{\sqrt{21} - 5} \\ &= \frac{5(5 - \sqrt{21})}{\sqrt{21} - 5}\tag1\end{align}$$.
$$\because \forall (x, y), \ \frac{x - y}{y - x} = \frac{x - y}{-(x - y)} = -1,$$
$$(1) \iff \frac{5(5 - \sqrt{21})}{\sqrt{21} - 5} = 5\times \frac{5 - \sqrt{21}}{\sqrt{21} - 5} = 5\times (-1) = -5$$
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