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}
$