Complex Number equation: $z+2\bar{z}= |\bar{z}+3|$

Question

Solve $z+2\bar{z}= |\bar{z}+3|$.

I'm new to complex numbers and need help solving this equation. Appreciate the assistance. Thanks.

**Edit: I've understood it now, i have to compare the real and imaginary parts. Thanks everyone, have a great day ahead!

Answer 1

HINT

Since

$$z+2\bar z=|\bar z+3|$$

and $z+\bar z=2Re(z)$ we have that $z=\bar z=x$ is real then it reduces to solve

$$3x=|x+3|$$

Answer 2

Let $z=x+iy$ then $$(x+iy)+(2x-2iy)=|x-iy+3|$$ $$3x-iy=|(x+3)-iy|$$ shows $$3x=|(x+3)-iy|~~~~~\text{and}~~~~~-iy=0$$ can you proceed?

Answer 3

Note that $z+2\overline z$ must be real, which is only possible if $z$ is real !

Then $$3x=|x+3|$$

and $x$ must be positive.

Finally,

$$3x=x+3.$$