Find the value
\[ a = 5 + 2\sqrt{6} \]
Solution:
\[ \frac{1}{a} = \frac{1}{5 + 2\sqrt{6}} \times \frac{5 – 2\sqrt{6}}{5 – 2\sqrt{6}} \]
\[ = \frac{5 – 2\sqrt{6}}{25 – 24} = 5 – 2\sqrt{6} \]
\[ a + \frac{1}{a} = (5 + 2\sqrt{6}) + (5 – 2\sqrt{6}) \]
\[ = 10 \]