First Principles AI

Solution

\begin{aligned} & \boxed{\text{Divide both sides by 2:}} \\ \\ & z^2 - \frac{3}{2}z + 2 = 0 \\ \\ & \boxed{\text{Isolate the constant term:}} \\ \\ & z^2 - \frac{3}{2}z = -2 \\ \\ & \boxed{\text{Add } \left(\frac{-3/2}{2}\right)^2 = \left(-\frac{3}{4}\right)^2 = \frac{9}{16} \text{ to both sides:}} \\ \\ & z^2 - \frac{3}{2}z + \frac{9}{16} = -2 + \frac{9}{16} \\ \\ & \left(z - \frac{3}{4}\right)^2 = -\frac{32}{16} + \frac{9}{16} = -\frac{23}{16} \\ \\ & \boxed{\text{Take square root:}} \\ \\ & z - \frac{3}{4} = \pm \sqrt{-\frac{23}{16}} = \pm \frac{\sqrt{23}}{4}i \\ \\ & z = \frac{3}{4} \pm \frac{\sqrt{23}}{4}i = \frac{3 \pm i\sqrt{23}}{4} \end{aligned}