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

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


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\begin{equation}
\begin{aligned}

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