Question

x + $\frac{1}{x}$ = 7
x - $\frac{1}{x}$ = ?

Answer 1

$x+\dfrac{1}{x}=7 \Big|^2 \Rightarrow x^2+2\cdot x\cdot \dfrac{1}{x}+\dfrac{1}{x^2}=49 \Rightarrow \\ \\ \Rightarrow x^2+2+\dfrac{1}{x^2} = 49 \Rightarrow x^2+\dfrac{1}{x^2} = 47\\ \\\\ x-\dfrac{1}{x}= t\Big|^2 \Rightarrow x^2-2+\dfrac{1}{x^2} = t^2 \Rightarrow\\ \\ \Rightarrow 47-2 = t^2 \Rightarrow t^2 = 45 \Rightarrow t = \pm \sqrt{45} \Rightarrow \\ \\ \Rightarrow t = \pm 3\sqrt{5} \Rightarrow \boxed{x-\dfrac{1}{x}=\pm 3\sqrt{5}}$