Question

If $x=4+\sqrt{15},$ then find $x+\frac{1}{x}$ is equal to

• A. $8$
• B. $4$
• C. $10$
• D. $6$

Answer 1

The correct option is A $8$

We are given that $x=4+\sqrt{15}$, so we first find $\frac{1}{x}$ as shown below:

$\frac{1}{x}=\frac{1}{4+\sqrt{15}}$
$=\frac{1}{4+\sqrt{15}}\times \frac{4-\sqrt{15}}{4-\sqrt{15}}\left(Rationalising\right)$
$=\frac{4-\sqrt{15}}{(4+\sqrt{15})(4-\sqrt{15})}$

$=\frac{4-\sqrt{15}}{4^2-(\sqrt{15}{)}^2}(\because (a+b)(a-b)=a^2-b^2)$

$=\frac{4-\sqrt{15}}{16-15}$
$=\frac{4-\sqrt{15}}{1}$
$=4-\sqrt{15}$

Therefore, $\frac{1}{x}=4-\sqrt{15}$

Now, let us find $x+\frac{1}{x}$as follows:

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

Hence, $x+\frac{1}{x}=8$.