Answer
The solutions are $x=-\sqrt{7}$ and $x=\sqrt{7}$
Work Step by Step
$\sqrt{11-x^2}-\dfrac{2}{\sqrt{11-x^2}}=1\hspace{0.7cm}{\color{blue}{\text{Given equation}}}$
$\Rightarrow \sqrt{11-x^2}\times (\sqrt{11-x^2}-\dfrac{2}{\sqrt{11-x^2}})=\sqrt{11-x^2}$
$\Rightarrow (\sqrt{11-x^2})^2-2=\sqrt{11-x^2}$
$\Rightarrow (\sqrt{11-x^2})^2-2-\sqrt{11-x^2}=0$
$\Rightarrow (\sqrt{11-x^2})^2-\sqrt{11-x^2}-2=0$
$\Rightarrow \bigg(\sqrt{11-x^2}-2\bigg)\bigg(\sqrt{11-x^2}+1\bigg)=0$
$\Rightarrow\sqrt{11-x^2}-2=0\Rightarrow \sqrt{11-x^2}=2\Rightarrow 11-x^2=4$
$\Rightarrow -x^2=4-11\Rightarrow-x^2=-7\Rightarrow x^2=7\Rightarrow x=\pm\sqrt{7}$
$\text{or }\sqrt{11-x^2}+1=0\Rightarrow \sqrt{11-x^2}=-1 \hspace{0.3cm}{\color{red}{reject}}$
The solutions are $x=-\sqrt{7}$ and $x=\sqrt{7}$
