College Algebra, Chapter 1, 1.4, Section 1.4, Problem 68

Find all solutions of the equation $z + 4 + \frac{12}{z} = 0$ and express them in the form $a + bi$.


$
\begin{equation}
\begin{aligned}

z + 4 + \frac{12}{z} =& 0
&& \text{Given}
\\
\\
z^2 + 4z + 12 =& 0
&& \text{Multiply both sides of the equation by } z
\\
\\
z^2 + 4z =& -12
&& \text{Subtract } 12
\\
\\
z^2 + 4z + 4 =& -12 + 4
&& \text{Complete the square: add } \left( \frac{4}{2} \right)^2 = 4
\\
\\
(z + 2)^2 =& -8
&& \text{Perfect square}
\\
\\
z + 2 =& \pm \sqrt{-8}
&& \text{Take the square root}
\\
\\
z + 2 =& \pm \sqrt{8i^2}
&& \text{Recall that } i^2 = -1
\\
\\
z =& -2 \pm 2 \sqrt{2} i
&& \text{Subtract } 2
\\
\\
(z + (2 + 2 \sqrt{2} i))& (z + (2 - 2 \sqrt{2} i)) = 0
&&


\end{aligned}
\end{equation}
$