(v) 2z2+6z+9=0 2z^2 + 6z + 9 = 02z2+6z+9=0 Solution Divide both sides by 2:z2+3z+92=0Isolate the constant term:z2+3z=−92Add (32)2=94 to both sides:z2+3z+94=−92+94(z+32)2=−184+94=−94Take square root:z+32=±−94=±32iz=−32±32i=−3±3i2\begin{aligned} & \boxed{\text{Divide both sides by 2:}} \\ \\ & z^2 + 3z + \frac{9}{2} = 0 \\ \\ & \boxed{\text{Isolate the constant term:}} \\ \\ & z^2 + 3z = -\frac{9}{2} \\ \\ & \boxed{\text{Add } \left(\frac{3}{2}\right)^2 = \frac{9}{4} \text{ to both sides:}} \\ \\ & z^2 + 3z + \frac{9}{4} = -\frac{9}{2} + \frac{9}{4} \\ \\ & \left(z + \frac{3}{2}\right)^2 = -\frac{18}{4} + \frac{9}{4} = -\frac{9}{4} \\ \\ & \boxed{\text{Take square root:}} \\ \\ & z + \frac{3}{2} = \pm \sqrt{-\frac{9}{4}} = \pm \frac{3}{2}i \\ \\ & z = -\frac{3}{2} \pm \frac{3}{2}i = \frac{-3 \pm 3i}{2} \end{aligned}Divide both sides by 2:z2+3z+29=0Isolate the constant term:z2+3z=−29Add (23)2=49 to both sides:z2+3z+49=−29+49(z+23)2=−418+49=−49Take square root:z+23=±−49=±23iz=−23±23i=2−3±3i