Question

Solve the equation $z+\sqrt{2}|z+1|+i=0$ for complex value of z.

Answer 1

Let the required complex number be z=x+iy. Then,

$z+\sqrt{2}|z+1|+i=0$

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

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

$\Rightarrow x+\sqrt{2}\left[\sqrt{(x+1{)}^2+y^2}\right]=0$ and $y+1=0$ [equating real parts and imaginary parts separately on both sides]

$\Rightarrow y=-1$ and $\sqrt{2}\left[\sqrt{(x+1{)}^2+(-1{)}^2}\right]=(-x)$

$\Rightarrow y=-1$ and $\sqrt{2}.\sqrt{x^2+2x+2}=(-x)$

$\Rightarrow y=-1and2(x^2+2x+2)=x^2$

$\Rightarrow y=-1$ and $x^2+4x+4=0\Rightarrow (x+2{)}^2=0$ and y=-1

$\Rightarrow x+2=0$ and $y=-1\Rightarrow x=-2$ and $y=-1$.

Hence, the required complex number is z=(-2-i).