Find the value
\[ x = 3 – \sqrt{8} \]
Solution:
\[ \frac{1}{x} = \frac{1}{3 – \sqrt{8}} \times \frac{3 + \sqrt{8}}{3 + \sqrt{8}} = 3 + \sqrt{8} \]
\[ x + \frac{1}{x} = (3 – \sqrt{8}) + (3 + \sqrt{8}) = 6 \]
\[ (x – \frac{1}{x})^2 = (x + \frac{1}{x})^2 – 4 \]
\[ = 6^2 – 4 = 36 – 4 = 32 \]