# find all z values

• Nov 1st 2009, 09:47 AM
metlx
find all z values
find all values for z for the following equation:

$\displaystyle z^2 + 2iRe(z) = |z|$

I tried
$\displaystyle |z|^2 \cdot (cos 2\phi + i \cdot sin 2\phi) + 2i \cdot a = |z|$

$\displaystyle |z| \cdot (cos 2\phi + i \cdot sin 2\phi) + \frac{2i \cdot a}{|z|} = 1$

what do i do now? can someone show me how to continue (or solve it if possible)?
• Nov 1st 2009, 06:33 PM
tonio
Quote:

Originally Posted by metlx
find all values for z for the following equation:

$\displaystyle z^2 + 2iRe(z) = |z|$

I tried
$\displaystyle |z|^2 \cdot (cos 2\phi + i \cdot sin 2\phi) + 2i \cdot a = |z|$

$\displaystyle |z| \cdot (cos 2\phi + i \cdot sin 2\phi) + \frac{2i \cdot a}{|z|} = 1$

what do i do now? can someone show me how to continue (or solve it if possible)?

Put z = x + yi, so your equation is:

$\displaystyle x^2-y^2+2x(y+1)i =\sqrt{x^2+y^2}$, and compare real and imaginary parts:

$\displaystyle ==\,\,x^2-y^2=\sqrt{x^2+y^2}$
$\displaystyle ==\,\,2x(y+1)=0\Longrightarrow x=0\,\,or\,\,y=-1$ and etc.

Tonio