Question

If $x=\left(4-\sqrt{15}\right)$, find the value of $\left(x+\frac{1}{x}\right)$.

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}: \\ x=4-\sqrt{15} \\ \frac{1}{x}=\frac{1}{4-\sqrt{15}}\times \frac{4+\sqrt{15}}{4+\sqrt{15}} \\ =\frac{4+\sqrt{15}}{16-15} \\ =4+\sqrt{15} \end{gathered}$$

$\therefore x+\frac{1}{x}=4-\sqrt{15}+4+\sqrt{15}=8$