Complex numbers

[Guest]

Hello.
I need help solving this question about complex numbers:
The solution to the equation|z|+z-18+6i=0 is:
a)6
b)8
c)18
d)12
e)10
Could you help me please? im quite confuse on how to do it

[Guest]

Write z as x+ iy. Then [tex]|z|= \sqrt{x^2+ y^2}[/tex] so [tex]|z|+ z- 18+ 6i= \sqrt{x^2+ y^2}+ x+ iy- 18+ 6i= 0[/tex].
Separating into real and imaginary parts, [tex]\sqrt{x^2+ y^2}+ x- 18= 0[/tex] and [tex]iy+ 6i= 0[/tex]. Solve those two equations for x and y.

I don't like squareroots so I would write the first equation as [tex]\sqrt{x^2+ y^2}= 18- x[/tex] and square both sides:
[tex]x^2+ y^2= 324-36x+ x^2[/tex]. The two "[tex]x^2[/tex] terms cancel so [tex]y^2= 324- 36x[/tex].
So we have [tex]y^2= 324- 36x[/tex] and [tex]y+ 6= 0[/tex].

Can you solve those equations for x and y?