(vi) 3z2−5z+7=0 3z^2 - 5z + 7 = 03z2−5z+7=0 Solution Divide both sides by 3:z2−53z+73=0Isolate the constant term:z2−53z=−73Add (−5/32)2=(−56)2=2536 to both sides:z2−53z+2536=−73+2536(z−56)2=−8436+2536=−5936Take square root:z−56=±−5936=±i596z=56±i596=5±i596\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}Divide both sides by 3:z2−35z+37=0Isolate the constant term:z2−35z=−37Add (2−5/3)2=(−65)2=3625 to both sides:z2−35z+3625=−37+3625(z−65)2=−3684+3625=−3659Take square root:z−65=±−3659=±6i59z=65±6i59=65±i59