Question

Solve:$x^2+\frac{1}{x^2}+2-2x-\frac{2}{x}$

Answer 1

$x^2+\frac{1}{x^2}+2-2x-\frac{2}{x}$

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

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

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

$=(x+\frac{1}{x})(\left(\sqrt{x}\right)+\frac{1}{{\left(\sqrt{x}\right)}^2}-2\sqrt{x}\times \frac{1}{\sqrt{x}})$

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