First Principles AI

Solution

\begin{aligned} & \boxed{\text{Divide both sides by 3:}} \\ \\ & z^2 - \frac{5}{3}z + \frac{7}{3} = 0 \\ \\ & \boxed{\text{Isolate the constant term:}} \\ \\ & z^2 - \frac{5}{3}z = -\frac{7}{3} \\ \\ & \boxed{\text{Add } \left(\frac{-5/3}{2}\right)^2 = \left(-\frac{5}{6}\right)^2 = \frac{25}{36} \text{ to both sides:}} \\ \\ & z^2 - \frac{5}{3}z + \frac{25}{36} = -\frac{7}{3} + \frac{25}{36} \\ \\ & \left(z - \frac{5}{6}\right)^2 = -\frac{84}{36} + \frac{25}{36} = -\frac{59}{36} \\ \\ & \boxed{\text{Take square root:}} \\ \\ & z - \frac{5}{6} = \pm \sqrt{-\frac{59}{36}} = \pm \frac{i\sqrt{59}}{6} \\ \\ & z = \frac{5}{6} \pm \frac{i\sqrt{59}}{6} = \frac{5 \pm i\sqrt{59}}{6} \end{aligned}