I have been practicing some complex numbers and came across this problem.
Find all the complex numbers z that satisfie the equation $z + |z| = 8+4i$
I have said that $z = x + yi$ which means $|z| = \sqrt{x^2+y^2}$ which makes the equation
$x + yi + \sqrt{x^2 +y^2} = 8 + 4i$ but I dont know how to go from here to a solution?
Do I just have to rearrange a little or am I completely barking up the wrong tree?
Best Answer
You may first identify real part and imaginary part:
$$x + yi + \sqrt{x^2 +y^2} = 8 + 4i\implies x+ \sqrt{x^2 +y^2}=8, yi=4i $$
$$ y=4, x+ \sqrt{x^2 +16}=8 $$
$$x=3, y=4 \implies z=3+4i$$