Question 3: \((vi)\) \(\quad\) If \(x=3+\sqrt{8}\) then find \(\left(x-\large{\frac{1}{x}}\right)^2 \)
Solution:
\[\begin{align*}
x&=3+\sqrt{8}\\
\frac{1}{x} &=\frac{1}{3+\sqrt{8}}\\
&=\frac{1}{3+\sqrt{8}} \times \frac{3-\sqrt{8}}{3-\sqrt{8}}\\
&=\frac{3-\sqrt{8}}{3^{2} -\left(\sqrt{8}\right)^{2}}\\
&=\frac{3-\sqrt{8}}{9-8}\\
&=3-\sqrt{8}\\
x-\frac{1}{x}&=3+\sqrt{8}-3+\sqrt{8}\\
&=2\sqrt{8}\\ \end{align*}\]
Taking Square on both sides
\[\begin{align*}\left( x-\frac{1}{x}\right)^{2} &=\left( 2\sqrt{8}\right)^{2}\\
&=4\times 8\\
&=32\\
\end{align*}\]