Find the complex number satisfying the equation z+2|(z+1)|+i=0 - Vidyadip

Question

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

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Answer

We have $\text{z}+\sqrt{2}|(\text{z}+1)|+\text{i}=0\ ....(\text{i})$
Putting $\text{z}=\text{x}+\text{iy},$ we get
$\text{x}+\text{iy}+\sqrt{2}|\text{x}+\text{iy}+1|+\text{i}=0$
$\Rightarrow\text{x}+\text{i}(1+\text{y})+\sqrt{2}\Big[\sqrt{(\text{x}+1)^2+\text{y}^2}\Big]=0$
$\Rightarrow\text{x}+\text{i}(1+\text{y})+\sqrt{2}\sqrt{(\text{x}^2+2\text{x}+1+\text{y}^2)}=0$
Comparing real and imaginary parts to zero, we get
$\text{x}+\sqrt{2}\sqrt{\text{x}^2+2\text{x}+1+\text{y}^2}=0$
And $\text{y}+1=0$
$\Rightarrow\text{y}=-1$
Putting y = -1 into (ii), we get
$\text{x}+\sqrt{2}\sqrt{\text{x}^2+2\text{x}+1+1}=0$
$\Rightarrow\sqrt{2}\sqrt{\text{x}^2+2\text{x}+2}=-\text{x}$
$\Rightarrow2\text{x}^2+4\text{x}+4=\text{x}^2$
$\Rightarrow\text{x}^2+4\text{x}+4=0$
$\Rightarrow(\text{x}+2)^2=0$
$\Rightarrow\text{x}=-2$
$\therefore\ \text{z}=\text{x}+\text{iy}=-2-\text{i}$

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