Let z_1 and z_2 be two complex numbers such that z_1+z_2=5 and z_1^3+z_2^3=20+15 i Then, […

Question

Let $z_1$ and $z_2$ be two complex numbers such that $z_1+z_2=5$ and $z_1^3+z_2^3=20+15 i$ Then, $$\left|z_1^4+z_2^4\right|$$ equals -

Accepted Answer

$$\begin{aligned}
& z_1+z_2=5 \
& z_1^3+z_2^3=20+15 i \
& z_1^3+z_2^3=\left(z_1+z_2\right)^3-3 z_1 z_2\left(z_1+z_2\right) \
& z_1^3+z_2^3=125-3 z_1 \cdot z_2(5) \
& \Rightarrow 20+15 i=125-15 z_1 z_2 \
& \Rightarrow 3 z_1 z_2=25-4-3 i \
& \Rightarrow 3 z_1 z_2=21-3 i \
& \Rightarrow z_1 \cdot z_2=7-i \
& \Rightarrow\left(z_1+z_2\right)^2=25 \
& \Rightarrow z_1^2+z_2^2=25-2(7-i) \
& \Rightarrow 11+2 i \
& \left(z_1^2+z_2^2\right)^2=121-4+44 i \
& \Rightarrow z_1^4+z_2^4+2(7-i)^2=117+44 i \
& \Rightarrow z_1^4+z_2^4=117+44 i-2(49-1-14 i) \
& \Rightarrow\left|z_1^4+z_2^4\right|=75
\end{aligned}$$

Solutions · 1