College Algebra, Chapter 1, 1.5, Section 1.5, Problem 70

Find all solutions, real and complex of the equation $\displaystyle 1 - \sqrt{x^2 + 7} = 6 - x^2$


$\begin{equation} \begin{aligned} 1 - \sqrt{x^2 + 7} =& 6 - x^2 && \text{Given} \\ \\ \sqrt{x^2 + 7} =& 5 - x^2 && \text{Subtract } 1 \\ \\ x^2 + 7 =& 25 - 10x^2 + x^4 && \text{Square both sides} \\ \\ x^4 - 11x^2 - 18 =& 0 && \text{Combine like terms} \\ \\ w^2 - 11w - 18 =& 0 && \text{Let } w = x^2 \\ \\ w^2 - 11w =& 18 && \text{Add } 18 \\ \\ w^2 - 11w + \frac{121}{4} =& 18 + \frac{121}{4} && \text{Complete the square: add } \left( \frac{-11}{2} \right)^2 = \frac{121}{4} \\ \\ \left( w - \frac{11}{2} \right)^2 =& \frac{193}{4} && \text{Perfect Square} \\ \\ w - \frac{11}{2} =& \pm \sqrt{\frac{193}{4}} && \text{Take the square root} \\ \\ w =& \frac{11}{2} \pm \frac{\sqrt{193}}{2} && \text{Add } \frac{11}{2} \text{ and simplify} \\ \\ w =& \frac{11 + \sqrt{193}}{2} \text{ and } w = \frac{11 - \sqrt{193}}{2} && \text{Solve for } w \\ \\ x^2 =& \pm \sqrt{\frac{11 + \sqrt{193}}{2}} \text{ and } \pm \sqrt{\frac{11 - \sqrt{193}}{2}} && \text{Solve for } x \end{aligned} \end{equation}$