Question

Find all complex numbers $z$ which satisfy the following equation
$z^2+\overline{z}=0$

• A. $1,w,2w$
• B. $1,w,w^2$
• C. Both a and b
• D. Not defined

Answer 1

Answer: (B)

$1,w,w^2$

$z$ be $x+iy$

$z^2=x^2+2ixy-y^2$

$\overline{z}=x-iy$

sum $=x^2+x-y^2+i(2xy-y)=0+0i$

so the answer is the solution to equation

$x^2+x-y^2=0$.................... ( i )

$(2x-1)y=0$.........................( ii )

first is pair of rectangular hyperbola $=>(x+0.5{)}^2-y^2=(0.5{)}^2$

second is pair of lines $x=\frac{1}{2}$ and $y=0$

for $y=0=>x=0,-1$

for $x=0.5=>y=\pm \left(\sqrt{0.75}\right)$

Hence $z=1,w,w^2$