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 $a + bi$.


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