College Algebra, Chapter 1, 1.4, Section 1.4, Problem 64

Find all solutions of the equation $x^2 - 3x + 3 = 0$ and express them in the form $a + bi$.


$
\begin{equation}
\begin{aligned}

x^2 - 3x + 3 =& 0
&& \text{Given}
\\
\\
x^2 - 3x =& -3
&& \text{Subtract } 3
\\
\\
x^2 - 3x + \frac{9}{4} =& -3 + \frac{9}{4}
&& \text{Complete the square: add } \left( \frac{-3}{2} \right)^2 = \frac{9}{4}
\\
\\
\left( x - \frac{3}{2} \right)^2 =& \frac{-3}{4}
&& \text{Perfect square}
\\
\\
x - \frac{3}{2} =& \pm \sqrt{\frac{-3}{4}}
&& \text{Take the square root}
\\
\\
x - \frac{3}{2} =& \pm \sqrt{\frac{3i^2}{4}}
&& \text{Recall that } i^2 = -1
\\
\\
x =& \frac{3}{2} \pm \frac{\sqrt{3}}{2} i
&& \text{Add } \frac{3}{2} \text{ and simplify}
\\
\\
\left( x - \left( \frac{3 + \sqrt{3} i}{2} \right) \right)&\left( x - \left( \frac{3 - \sqrt{3} i}{2} \right) \right) = 0
&&


\end{aligned}
\end{equation}
$