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

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


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