$\sqrt{x^2-\frac{7}{x^2}}=x-\sqrt{x-\frac{7}{x^2}}\Rightarrow x^2-\frac{7}{x^2}=x^2+x-\frac{7}{x^2}-2x\sqrt{x-\frac{7}{x^2}}$$\Rightarrow 1=2\sqrt{x-\frac{7}{x^2}}\Rightarrow 1=4(x-\frac{7}{x^2})\Leftrightarrow 4x^3-28-x^2=0\Leftrightarrow (x-2)(4x^2+7x+14)=0\Rightarrow x=2$