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  4. The complex number which satisfies the equation z+√2|z+1|+i=0 is

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Q. The complex number which satisfies the equation $z+\sqrt{2}|z+1|+i=0$ is

Complex Numbers and Quadratic Equations

Solution:

Since $z+\sqrt{2}|z+1|+i=0$
$\therefore x+i(y+1)+\sqrt{2}|x+i y+1|=0$
$\therefore y+1=0$
$(\because|x+i y+1|$ is real $)$
$\therefore y=-1$
$\therefore x+\sqrt{2}|x-i+1|=0$
$\Rightarrow x^{2}=2\left[(x+1)^{2}+1\right]$
$=2\left(x^{2}+2 x+2\right)$
$\therefore x^{2}+4 x+4=0 $
$ \Rightarrow (x+2)^{2}=0 $
or $ x=-2 $
$ \therefore z=-2-i.$