Question

If x - $\frac{1}{x}$ = $\sqrt{5}$ then find
(i) $2x^2$ - $\frac{2}{x^2}$ (ii) $3x$ + $\frac{3}{x}$

Answer 1

$\Rightarrow x-\frac{1}{x}=\sqrt{5}$

$\Rightarrow x^2+\frac{1}{x^2}-2=5$

$\Rightarrow x^2+\frac{1}{x^2}=7$

$\Rightarrow {(x+\frac{1}{x})}^2-2=7$

$\Rightarrow (x+\frac{1}{x})=3$

(i) $2x^2-\frac{2}{x^2}$

$\Rightarrow 2(x+\frac{1}{x})(x-\frac{1}{x})=2\times 3\times \sqrt{5}=6\sqrt{5}$

(ii)$3x+\frac{3}{x}$

$\Rightarrow 3(x+\frac{1}{x})=3\times 3=9$