Find the value
\[ x = 3 + \sqrt{8} \]
\[ \frac{1}{x} = \frac{1}{3 + \sqrt{8}} \times \frac{3 – \sqrt{8}}{3 – \sqrt{8}} = 3 – \sqrt{8} \]
\[ x + \frac{1}{x} = (3 + \sqrt{8}) + (3 – \sqrt{8}) = 6 \]
\[ x^2 + \frac{1}{x^2} = (x + \frac{1}{x})^2 – 2 \]
\[ = 6^2 – 2 = 36 – 2 = 34 \]