College Algebra, Chapter 1, 1.5, Section 1.5, Problem 68

Find all solutions, real and complex of the equation $\displaystyle x^6 + 9x^4 - 4x^2 - 36 = 0$


$
\begin{equation}
\begin{aligned}

x^6 + 9x^4 - 4x^2 - 36 =& 0
&& \text{Given}
\\
\\
(x^6 + 9x^4) - (4x^2 + 36) =& 0
&& \text{Group terms}
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x^4(x^2 + 9) - 4 (x^2 + 9) =& 0
&& \text{Factor out } x^4 \text{ and } 4
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(x^4 - 4)(x^2 + 9) =& 0
&& \text{Factor out } x^4 - 4
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x^4 4 =& 0 \text{ and } x^2 + 9 = 0
&& \text{Zero Product Property}
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x^4 =& 4 \text{ and } x^2 = -9
&& \text{Solve for } x
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x =& \pm \sqrt[4]{4} \text{ and } x = \pm \sqrt{-9}
&& \text{Recall that } i^2 = -1
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x =& \pm \sqrt[4]{4} \text{ and } x = \pm \sqrt{9i^2}
&& \text{Simplify}
\\
\\
x =& \pm \sqrt[4]{4} \text{ and } x = \pm 3i
&&

\end{aligned}
\end{equation}
$