College Algebra, Chapter 1, 1.3, Section 1.3, Problem 52

Find all real solutions of $\displaystyle 5x^2 - 7x + 5 = 0$.


$\begin{equation} \begin{aligned} 5x^2 - 7x + 5 =& 0 && \text{Given} \\ \\ 5x^2 - 7x =& -5 && \text{Subtract 5} \\ \\ x^2 - \frac{7x}{5} =& -1 && \text{Divide both sides by 5 to make the coefficient of $x^2$ equal to 1} \\ \\ x^2 - \frac{7x}{5} + \frac{49}{100} =& -1 + \frac{49}{100} && \text{Complete the square: add } \left( \frac{\displaystyle \frac{-7}{5}}{2} \right)^2 = \frac{49}{100} \\ \\ \left(x - \frac{7}{10} \right)^2 =& \frac{-51}{100} && \text{Perfect square} \\ \\ x - \frac{7}{10} =& \pm \sqrt{\frac{-51}{100}} && \text{Take the square root} \end{aligned} \end{equation}$


No real solution the discriminant $b^2 - 4ac < 0$