Find the value
\[ x = 2 + \sqrt{3} \]
Solution:
\[ \frac{1}{x} = \frac{1}{2 + \sqrt{3}} \times \frac{2 – \sqrt{3}}{2 – \sqrt{3}} \]
\[ = \frac{2 – \sqrt{3}}{4 – 3} = 2 – \sqrt{3} \]
\[ x + \frac{1}{x} = (2 + \sqrt{3}) + (2 – \sqrt{3}) \]
\[ = 4 \]
\[ x = 2 + \sqrt{3} \]
\[ \frac{1}{x} = \frac{1}{2 + \sqrt{3}} \times \frac{2 – \sqrt{3}}{2 – \sqrt{3}} \]
\[ = \frac{2 – \sqrt{3}}{4 – 3} = 2 – \sqrt{3} \]
\[ x + \frac{1}{x} = (2 + \sqrt{3}) + (2 – \sqrt{3}) \]
\[ = 4 \]