Question

If $x=2+\sqrt{3}\mathrm{then}\left(x+\frac{1}{x}\right)$ equals
(a) $-2\sqrt{3}$
(b) 2
(c) 4
(d) $4-2\sqrt{3}$

Answer 1

$$\begin{gathered} x=2+\sqrt{3} \\ \frac{1}{x}=\frac{1}{2+\sqrt{3}}=\frac{1}{2+\sqrt{3}}\times \frac{2-\sqrt{3}}{2-\sqrt{3}}=\frac{2-\sqrt{3}}{4-3}=2-\sqrt{3} \end{gathered}$$
$x+\frac{1}{x}=2+\sqrt{3}+2-\sqrt{3}=4$
Hence, the correct answer is option (c).