# 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 $-5$.

$$\frac{\sqrt{21}-5}{2}+\frac{2}{\sqrt{21}-5}=\frac{\sqrt{21}-5}{2}+\frac{2(\sqrt{21}+5)}{(\sqrt{21}-5)(\sqrt{21}+5)}=\frac{\sqrt{21}-5}{2}+\frac{2\sqrt{21}+10}{-4}=\frac{2\sqrt{21}-10}{4}-\frac{2\sqrt{21}+10}{4}=\frac{2\sqrt{21}-10-2\sqrt{21}-10}{4}=-5$$
• You don't need to use $4$ as the common denominator, since $2\sqrt{21}+10$ is obviously divisible by $2$. :) – Thomas Andrews Aug 24 '14 at 20:03
$$\frac{\sqrt{21}-5}2+\frac2{\sqrt{21}-5}=\frac{50-10\sqrt{21}}{2\sqrt{21}-10}=-5\frac{\sqrt{21}-5}{\sqrt{21}-5}$$
$$\frac{2}{\sqrt{21}-5}=\frac{2}{\sqrt{21}-5}\frac{\sqrt{21}+5}{\sqrt{21}+5}=\cdots=-\frac{\sqrt{21}+5}{2}$$
\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}. $$\because \forall (x, y), \ \frac{x - y}{y - x} = \frac{x - y}{-(x - y)} = -1,$$ $$(1) \iff \frac{5(5 - \sqrt{21})}{\sqrt{21} - 5} = 5\times \frac{5 - \sqrt{21}}{\sqrt{21} - 5} = 5\times (-1) = -5$$