First Principles AI

Solution

\begin{aligned} & \boxed{\text{Divide both sides by 3:}} \\ \\ & z^2 - 6z + \frac{50}{3} = 0 \\ \\ & \boxed{\text{Isolate the constant term:}} \\ \\ & z^2 - 6z = -\frac{50}{3} \\ \\ & \boxed{\text{Add } \left(\frac{-6}{2}\right)^2 = (-3)^2 = 9 \text{ to both sides:}} \\ \\ & z^2 - 6z + 9 = -\frac{50}{3} + 9 \\ \\ & (z - 3)^2 = -\frac{50}{3} + \frac{27}{3} = -\frac{23}{3} \\ \\ & \boxed{\text{Take square root:}} \\ \\ & z - 3 = \pm \sqrt{-\frac{23}{3}} = \pm i\sqrt{\frac{23}{3}} = \pm \frac{i\sqrt{69}}{3} \\ \\ & z = 3 \pm \frac{i\sqrt{69}}{3} \end{aligned}