# Complex numbers

Algebra

### Complex numbers

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

### Re: Complex numbers

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

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

Can you solve those equations for x and y?
Guest

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