College Algebra, Chapter 1, 1.3, Section 1.3, Problem 46

Find all real solutions of $\displaystyle z^2 + 5z + 3 = 0$.


$\begin{equation} \begin{aligned} z^2 + 5z + 3 =& 0 && \text{Given} \\ \\ z^2 + 5z =& -3 && \text{Subtract 3} \\ \\ z^2 + 5z + \frac{25}{4} =& -3 + \frac{25}{4} && \text{Complete the square: add } \left( \frac{5}{2} \right)^2 = \frac{25}{4} \\ \\ \left( z + \frac{5}{2} \right)^2 =& \frac{13}{4} && \text{Perfect square, take the LCD on the right side then simplify} \\ \\ z + \frac{5}{2} =& \pm \sqrt{\frac{13}{4}} && \text{Take the square root} \\ \\ z =& \frac{-5}{2} \pm \sqrt{\frac{13}{4}} && \text{Subtract } \frac{5}{2} \\ \\ z =& \frac{-5}{2} \pm \sqrt{\frac{13}{4}} && \text{Subtract } \frac{5}{2} \\ \\ z =& \frac{-5}{2} + \frac{\sqrt{13}}{2} \text{ and } z = \frac{-5}{2} - \frac{\sqrt{13}}{2} && \text{Solve for } z \\ \\ z =& \frac{-5 + \sqrt{13}q}{2} \text{ and } z = \frac{-5 - \sqrt{13}}{2} && \text{Simplify} \end{aligned} \end{equation}$