Find the value
\[ x = \sqrt{6} + \sqrt{5} \]
Solution:
\[ \frac{1}{x} = \frac{1}{\sqrt{6} + \sqrt{5}} \times \frac{\sqrt{6} – \sqrt{5}}{\sqrt{6} – \sqrt{5}} \]
\[ = \sqrt{6} – \sqrt{5} \]
\[ x + \frac{1}{x} = (\sqrt{6} + \sqrt{5}) + (\sqrt{6} – \sqrt{5}) = 2\sqrt{6} \]
\[ x^2 + \frac{1}{x^2} – 2 = (x + \frac{1}{x})^2 – 4 \]
\[ = (2\sqrt{6})^2 – 4 = 24 – 4 = 20 \]