Question 3: \((i)\) If \(x=3+\sqrt{8}\) then find \(x+\large{\frac{1}{x}} \)
Solution:
\[\begin{align*}
x&=3+\sqrt{8}\\
\frac{1}{x} &=\frac{1}{3+\sqrt{8}}\\
&=\frac{1}{3+\sqrt{8}} \times \frac{3-\sqrt{8}}{3-\sqrt{8}}\\
&=\frac{3-\sqrt{8}}{3^{2} -\left(\sqrt{8}\right)^{2}}\\
&=\frac{3-\sqrt{8}}{9-8}\\
&=3-\sqrt{8}\\
x+\frac{1}{x}&=3+\sqrt{8}+3-\sqrt{8}\\
&=6\\
\end{align*}\]