Question

Find the complex number satisfying the equation $z+\sqrt{2}|(z+1)|+i=0$

Answer 1

Given : $z+\sqrt{2}|(z+1)|+i=0$

Let $z=x+iy$

$\Rightarrow x+iy+\sqrt{2}|x+1+iy|+i=0$

$\Rightarrow x+i(y+1)+\sqrt{2}\sqrt{(x+1{)}^2+y^2}=0$

$\Rightarrow x+i(y+1)+\sqrt{2}\sqrt{x^2+y^2+2x+1}=0$

Now, comparing the real and imaginary parts, we get

$x+\sqrt{2}\sqrt{x^2+y^2+2x+1}=0$ ⋯(1)

$y+1=0$

$\Rightarrow y=-1$ ⋯(2)

Using equations (1) and (2), we get

$\Rightarrow x+\sqrt{2}\sqrt{x^2+2x+2}=0$

$\Rightarrow x=-\sqrt{2}\sqrt{x^2+2x+2}$

Squaring on both sides, we get

$\Rightarrow x^2=2(x^2+2x+2)$

$\Rightarrow x^2+4x+4=0$

$\Rightarrow (x+2{)}^2=0$

$\Rightarrow x=-2$

Hence, $z=x+iy=-2-i$