Question

If $x^2+\frac{1}{x^2}=7$, find the values of $3x^2-\frac{3}{x^2}$.

• A. $\pm 9\sqrt{2}$
• B. $\pm 9\sqrt{8}$
• C. $\pm 9\sqrt{5}$
• D. $\pm 9\sqrt{3}$

Answer 1

The correct option is C $\pm 9\sqrt{5}$

$x^2+\frac{1}{x^2}=7$
Now,
$(x+\frac{1}{x}{)}^2=x^2+\frac{1}{x^2}+2(x\left)\right(\frac{1}{x})$
$=7+2$
$=>(x+\frac{1}{x})=\sqrt{9}$
$=>(x+\frac{1}{x})=\pm 3$
Now,
$(x-\frac{1}{x}{)}^2=x^2+\frac{1}{x^2}-2(x\left)\right(\frac{1}{x})$
$=7-2$
$=5$
$(x-\frac{1}{x})=\sqrt{5}$
Now,
$3x^2-\frac{3}{x^2}=3(x^2-\frac{1}{x^2})$
$=3(x+\frac{1}{x})(x-\frac{1}{x})$
$=3(\pm 3)\left(\sqrt{5}\right)$
$=\pm 9\sqrt{5}$