College Algebra, Chapter 1, 1.3, Section 1.3, Problem 22

Solve $\displaystyle x^2 - 5x + 1 = 0$ by completing the square.


$
\begin{equation}
\begin{aligned}

x^2 - 5x + 1 =& 0
&& \text{Given}
\\
\\
x^2 - 5x =& -1
&& \text{Subtract 1}
\\
\\
x^2 - 5x + \frac{25}{4} =& -1 + \frac{25}{4}
&& \text{Complete the square: add } \left( \frac{-5}{2} \right)^2 = \frac{25}{4}
\\
\\
\left( x - \frac{5}{2} \right)^2 =& \frac{21}{4}
&& \text{Perfect square}
\\
\\
x - \frac{5}{2} =& \pm \sqrt{\frac{21}{4}}
&& \text{Take square root}
\\
\\
x =& \pm \frac{\sqrt{21}}{2} + \frac{5}{2}
&& \text{Add } \frac{5}{2}
\\
\\
x =& \frac{5 + \sqrt{21}}{2} \text{ and } x = \frac{5 - \sqrt{21}}{2}
&& \text{Solve for } x

\end{aligned}
\end{equation}
$