College Algebra, Chapter 1, 1.4, Section 1.4, Problem 60

Find all solutions of the equation $x^2 + 2x + 2 = 0$ and express them in the form $a + bi$.


$\begin{equation} \begin{aligned} x^2 + 2x + 2 =& 0 && \text{Given} \\ \\ x^2 + 2x =& -2 && \text{Subtract } 2 \\ \\ x^2 + 2x + 1 =& -2 + 1 && \text{Complete the square: add } \left( \frac{2}{2} \right)^2 = 1 \\ \\ (x + 1)^2 =& -1 && \text{Perfect square} \\ \\ x + 1 =& \pm \sqrt{-1} && \text{Take the square root} \\ \\ x + 1 =& \pm \sqrt{i^2} && \text{Recall that } i^2 = -1 \\ \\ x + 1 =& \pm i && \text{Subtract } 1 \\ \\ x =& -1 \pm i && \\ \\ (x + (1 + i))(x + (1 - i)) =& 0 && \end{aligned} \end{equation}$