College Algebra, Chapter 1, 1.4, Section 1.4, Problem 66

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


$
\begin{equation}
\begin{aligned}

2x^2 + 3 =& 2x
&& \text{Given}
\\
\\
2x^2 - 2x + 3 =& 0
&& \text{Subtract } 2x \text{ to both sides of the equation}
\\
\\
x^2 - x + \frac{3}{2} =& 0
&& \text{Subtract } \frac{3}{2}
\\
\\
x^2 - x + \frac{1}{4} =& \frac{-3}{2} + \frac{1}{4}
&& \text{Complete the square: add } \left( \frac{-1}{2} \right)^2 = \frac{1}{4}
\\
\\
\left( x - \frac{1}{2} \right)^2 =& \frac{-5}{4}
&& \text{Perfect square}
\\
\\
x - \frac{1}{2} =& \pm \sqrt{\frac{-5}{4}}
&& \text{Take the square root}
\\
\\
x - \frac{1}{2} =& \pm \sqrt{\frac{5i^2}{4}}
&& \text{Recall that } i^2 = -1
\\
\\
x =& \frac{1}{2} \pm \frac{\sqrt{5}}{2} i
&& \text{Add } \frac{1}{2} \text{ and simplify}
\\
\\
\left( x - \left( \frac{1 + \sqrt{5}}{2} \right) \right) & \left( x - \left( \frac{1 - \sqrt{5}}{2} \right) \right) = 0
&&


\end{aligned}
\end{equation}
$