Complex Numbers
System of equations via cube of complex sum
MJAT_TS2_P2
Grade 12
Question:
Let $z_1$ and $z_2$ be two complex numbers satisfying:
$$z_1^3 - 3z_1z_2^2 = 2, \qquad 3z_1^2z_2 - z_2^3 = 11$$
Find the value of $z_1^2 + z_2^2$.
Step-by-Step Solution
Key Concept: Recognise: $(z_1+iz_2)^3 = z_1^3-3z_1z_2^2 + i(3z_1^2z_2-z_2^3) = 2+11i$. Similarly $(z_1-iz_2)^3=2-11i$. Then $(z_1+iz_2)^3(z_1-iz_2)^3 = (z_1^2+z_2^2)^3 = (2+11i)(2-11i) = 4+121 = 125$.
$(z_1^2+z_2^2)^3=125 \Rightarrow z_1^2+z_2^2=\mathbf{5}$.
Correct Answer: 5