Question:
If $x=2+\sqrt{3}$ then $\left(x+\frac{1}{x}\right)$ equals
(a) $-2 \sqrt{3}$
(b) 2
(c) 4
(d) $4-2 \sqrt{3}$
Solution:
$x=2+\sqrt{3}$
$\frac{1}{x}=\frac{1}{2+\sqrt{3}}=\frac{1}{2+\sqrt{3}} \times \frac{2-\sqrt{3}}{2-\sqrt{3}}=\frac{2-\sqrt{3}}{4-3}=2-\sqrt{3}$
$x+\frac{1}{x}=2+\sqrt{3}+2-\sqrt{3}=4$
Hence, the correct answer is option (c).