Let S = z C : z^2 + 6 ,iz - 3 = 0 . Then _ z S z^8 is equal to :

Step-by-step solution

Given equation is z^2 + 6 iz - 3 = 0 . Rearranging the terms, we get z^2 - 3 = - 6 iz . Squaring both sides, we obtain: (z^2 - 3)^2 = (- 6 iz)^2 z^4 - 6z^2 + 9 = -6z^2 z^4 + 9 = 0 z^4 = -9 Squaring again, we get: (z^4)^2 = (-9)^2 z^8 = 81 Since the given equation is a quadratic in z , it has two roots, say z₁ and z₂ . The discriminant of the quadratic equation is = ( 6 i)^2 - 4(1)(-3) = -6 + 12 = 6 0 , which means the roots are distinct. Thus, the set S contains two distinct elements z₁ and z₂ . For each root, z^8