Find the value
If \[ x = \frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} – \sqrt{3}} \quad \text{and} \quad y = \frac{\sqrt{5} – \sqrt{3}}{\sqrt{5} + \sqrt{3}}, \] then \[ x + y + xy = \ ? \]
Solution:
\[ x = \frac{(\sqrt{5} + \sqrt{3})^2}{5 – 3} = \frac{5 + 3 + 2\sqrt{15}}{2} = 4 + \sqrt{15} \]
\[ y = \frac{(\sqrt{5} – \sqrt{3})^2}{5 – 3} = \frac{5 + 3 – 2\sqrt{15}}{2} = 4 – \sqrt{15} \]
\[ x + y = 8, \quad xy = 1 \]
\[ x + y + xy = 8 + 1 = 9 \]