Let S= z C : z^2+4z+16=0 . Then _ z S |z+ 3 i|^2 is equal to:

Step-by-step solution

Given the equation z^2 + 4z + 16 = 0 , we find its roots using the quadratic formula: z = -4 16 - 64 2 = -4 -48 2 = -2 2 3 i The set S contains two complex numbers: z₁ = -2 + 2 3 i and z₂ = -2 - 2 3 i . We need to evaluate the sum _ z S |z + 3 i|^2 , which is |z₁ + 3 i|^2 + |z₂ + 3 i|^2 . For z₁ = -2 + 2 3 i : |z₁ + 3 i|^2 = |-2 + 2 3 i + 3 i|^2 = |-2 + 3 3 i|^2 |-2 + 3 3 i|^2 = (-2)^2 + (3 3 )^2 = 4 + 27 = 31 For z₂ = -2 - 2 3 i : |z₂ + 3 i|^2 = |-2 - 2 3 i + 3 i|^2 = |-2 - 3 i|^2 |-2 - 3 i|^2 = (-2)^2 + (- 3 )^2