Question

If $x=3+2\sqrt{2}$, check whether $x+\frac{1}{x}$ is rational or irrational.

Answer 1

$x=3+2\sqrt{2}.....\left(1\right)$
$$\begin{gathered} \Rightarrow \frac{1}{x}=\frac{1}{3+2\sqrt{2}} \\ \Rightarrow \frac{1}{x}=\frac{1}{3+2\sqrt{2}}\times \frac{3-2\sqrt{2}}{3-2\sqrt{2}} \\ \Rightarrow \frac{1}{x}=\frac{3-2\sqrt{2}}{3^2-{\left(2\sqrt{2}\right)}^2} \end{gathered}$$
$$\begin{gathered} \Rightarrow \frac{1}{x}=\frac{3-2\sqrt{2}}{9-8} \\ \Rightarrow \frac{1}{x}=3-2\sqrt{2}.....\left(2\right) \end{gathered}$$
Adding (1) and (2), we get
$x+\frac{1}{x}=3+2\sqrt{2}+3-2\sqrt{2}=6$, which is a rational number
Thus, $x+\frac{1}{x}$ is rational.