Question

If $a=2+\sqrt{3}$ and $b=2-\sqrt{3}$, then $\frac{1}{a^2}-\frac{1}{b^2}$ is equal to

• A. $14$
• B. $-14$
• C. $8\sqrt{3}$
• D. $-8\sqrt{3}$

Answer 1

The correct option is D $-8\sqrt{3}$

$a=2+\sqrt{3}$
$b=2-\sqrt{3}$
$\frac{1}{a}=2-\sqrt{3}and\frac{1}{b}=2+\sqrt{3}$
$\therefore \frac{1}{a^2}-\frac{1}{b^2}=\left(\frac{1}{a}+\frac{1}{b}\right)\left(\frac{1}{a}-\frac{1}{b}\right)$
$=(2-\sqrt{3}+2+\sqrt{3})(2-\sqrt{3}-2-\sqrt{3})=4(-2\sqrt{3})=-8\sqrt{3}$