Question

find the complex number z that satisfies the equation, z+√2|z+1| + i =0

Answer 1

this may help u..........

Answer 2

Therefore the complex number  $z=-2-i$

Step-by-step explanation:

Given;

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

∴$x+iy+\sqrt{2} \left |x+iy+1 \right |+i=0$  where $z=x+iy$__equation-1

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

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

Now,

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

⇒$\sqrt{2(x+1)^{2} +2y^{2} }=-x$

⇒$2(x^{2} +2x+1)+2y^{2} =x^{2}$

⇒$2x^{2} +4x+2+2-x^{2} =0$  where $y=-1$

⇒$x^{2} +4x+4=0$

⇒$(x+2)^{2} =0$

⇒$x+2=0$

∴$x=-2$

Plug $x$ and $y$ value in equation-1;

$z=-2-i$

So the complex number  $z=-2-i$