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

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


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