I have to tackle the following question. My thoughts so far are below it.
Given that $z$ is the complex number $x+iy$ and that $|z|+z=6-2i$, find the value of $x$ and the value of $y$.
What I only know is that $|z|=r=\sqrt{x^2+y^2}$
So how should I continue to solve this problem?
On
WLOG $z=r(\cos2t+i\sin2t)$ where $r >0,t$ are real and $0\le2t<2\pi$
Equate the Real & the imaginary parts
$r(1+\cos2t)=6$ and $r\sin2t=-2$
On division $\tan t=-\dfrac26$
Use https://www.cut-the-knot.org/arithmetic/algebra/WeierstrassSubstitution.shtml
to find $\cos2t,\sin2t$
Then can you find $r?$
The key realization is that $|z|$ is always a real number.
So if we subtract $|z|$ from both sides, we see, grouping real and imaginary parts together,
$$z = (6 - |z|) + (-2)i$$
So, since $z = x+iy$ and the imaginary part of both sides must be equal, we can conclude $y=-2$. This also means $|z| = \sqrt{x^2 + (-2)^2} = \sqrt{x^2+4}$.
So, substituting in everything we know back into our original equation.
$$\sqrt{x^2+4} + x -2i = 6 - 2i$$
Add $2i$ to both sides and we get
$$\sqrt{x^2+4} + x = 6$$
This is a relatively simple algebraic equation to solve; once you do, you will have your $x$. (Be sure to check the $x$ you get: solving the equation above will introduce extraneous solutions!)