(i) 2z2−3z+4=02z^2 - 3z + 4 = 02z2−3z+4=0 Solution Divide both sides by 2:z2−32z+2=0Isolate the constant term:z2−32z=−2Add (−3/22)2=(−34)2=916 to both sides:z2−32z+916=−2+916(z−34)2=−3216+916=−2316Take square root:z−34=±−2316=±234iz=34±234i=3±i234\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}Divide both sides by 2:z2−23z+2=0Isolate the constant term:z2−23z=−2Add (2−3/2)2=(−43)2=169 to both sides:z2−23z+169=−2+169(z−43)2=−1632+169=−1623Take square root:z−43=±−1623=±423iz=43±423i=43±i23