College Algebra, Chapter 1, 1.4, Section 1.4, Problem 72

Find all solutions of the equation $\displaystyle x^2 + \frac{1}{2} x + 1 = 0$ and express them in the form $a + bi$.


$
\begin{equation}
\begin{aligned}

x^2 + \frac{1}{2} x + 1 =& 0
&& \text{Given}
\\
\\
x^2 + \frac{1}{2x} =& -1
&& \text{Subtract } 1
\\
\\
x^2 + \frac{1}{2}x + \frac{1}{16} =& -1 + \frac{1}{16}
&& \text{Complete the square: add } \left( \frac{\displaystyle \frac{1}{2}}{2} \right)^2 = \frac{1}{16}
\\
\\
\left(x + \frac{1}{4} \right)^2 =& \frac{-15}{16}
&& \text{Perfect square}
\\
\\
x + \frac{1}{4} =& \pm \sqrt{\frac{-15}{16}}
&& \text{Take the square root}
\\
\\
x + \frac{1}{4} =& \pm \frac{15 i^2}{16}
&& \text{Recall that } i^2 = -1
\\
\\
x =& \frac{-1}{4} \pm \frac{\sqrt{15}}{4} i
&& \text{Subtract } \frac{1}{4} \text{ and simplify}
\\
\\
\left( x + \left( \frac{1 + \sqrt{15} i}{4} \right) \right)& \left( x + \left( \frac{1 - \sqrt{15} i}{4} \right) \right) = 0
&&

\end{aligned}
\end{equation}
$