College Algebra, Chapter 1, 1.5, Section 1.5, Problem 58

Find all real solutions of the equation $\displaystyle \sqrt{11 - x^2} - \frac{2}{\sqrt{11 - x^2}} = 1$


$
\begin{equation}
\begin{aligned}

\sqrt{11 - x^2} - \frac{2}{\sqrt{11 - x^2}} =& 1
&& \text{Given}
\\
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11 - x^2 - 2 =& \sqrt{11 - x^2}
&& \text{Multiply both sides by } \sqrt{11 - x^2}
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(9 -x^2)^2 =& (\sqrt{11 - x^2})^2
&& \text{Square both sides}
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81 - 18x^2 + x^4 =& 11 -x^2
&& \text{Combine like terms}
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x^4 - 17x^2 + 70 =& 0
&& \text{Let } w = x^2
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w^2 - 17w + 70 =& 0
&& \text{Factor}
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(w - 7)(w - 10) =& 0
&& \text{Zero Product Property}
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w - 7 =& 0 \text{ and } w - 10 = 0
&& \text{Solve for } w
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w =& 7 \text{ and } w = 10
&& \text{Substitute } w = x^2
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x^2 =& 7 \text{ and } x^2 = 10
&& \text{Solve for } x
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x =& \pm \sqrt{7} \text{ and } x = \pm \sqrt{10}
&&
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x =& \pm \sqrt{7}
&& \text{The solutions that satisfy the equation}

\end{aligned}
\end{equation}
$