Find the value
\[ x = \frac{2}{\sqrt{3} – \sqrt{5}}, \quad y = \frac{2}{\sqrt{3} + \sqrt{5}} \]
Solution:
\[ x + y = \frac{2}{\sqrt{3} – \sqrt{5}} + \frac{2}{\sqrt{3} + \sqrt{5}} \]
\[ = \frac{2(\sqrt{3} + \sqrt{5}) + 2(\sqrt{3} – \sqrt{5})}{(\sqrt{3})^2 – (\sqrt{5})^2} \]
\[ = \frac{2\sqrt{3} + 2\sqrt{5} + 2\sqrt{3} – 2\sqrt{5}}{3 – 5} \]
\[ = \frac{4\sqrt{3}}{-2} \]
\[ = -2\sqrt{3} \]