College Algebra, Chapter 1, 1.5, Section 1.5, Problem 44

Find all real solutions of the equation $\displaystyle x^8 + 15x^4 = 16$


$
\begin{equation}
\begin{aligned}

x^8 + 15x^4 =& 16
&& \text{Given}
\\
\\
x^8 + 15x^4 - 16 =& 0
&& \text{Subtract } 16
\\
\\
w^2 + 15w - 16 =& 0
&& \text{Let } w = x^4
\\
\\
(w + 16)(w - 1)=& 0
&& \text{Factor}
\\
\\
w + 16 =& 0 \text{ and } w - 1 = 0
&& \text{Zero Product Property}
\\
\\
w =& -16 \text{ and } w = 1
&& \text{Solve for } w
\\
\\
x^4 =& -16 \text{ and } x^4 = 1
&& \text{Substitute } w = x^4
\\
\\
x =& \sqrt[4]{-16} \text{ and } x = \pm 1
&& \text{Solve for } x
\\
\\
x =& \pm 1
&& \text{The only real solution to the equation } x^8 + 15x^4 = 16

\end{aligned}
\end{equation}
$