Question

Solve the equation $z^2+z\left|z\right|+\left|z^2\right|=0.$

• A. $0$
• B. $k\omega$,$k(\omega {)}^2$, where k is non negative number
• C. $0,k\omega$,$k(\omega {)}^2$, where k is non negative number
• D. $0,-k\omega$$,-k(\omega {)}^2$, where k is non negative number

Answer 1

Answer: (C)

$0,k\omega$,$k(\omega {)}^2$, where k is non negative number

Clearly $z=0$ is one obvious solution.
When $z\ne 0$ then dividing by ${\left|z\right|}^2$ the equation can be written as $t^2+t+1=0$
where $t=\frac{z}{\left|z\right|}$
$\therefore t=\frac{-1\pm i\sqrt{3}}{2}$
or $\frac{z}{\left|z\right|}=\omega ,{\omega }^2$
$\therefore z=k\omega ,k{\omega }^2$ where $k=\left|z\right|$ is non-negative real number.
Ans: C