1. Simple equation problem

2. $y^2$ cannot be a negative number.
therefore the answer is $\oslash$

$y^2 = -\frac{1}{4}$
$y = \sqrt{-\frac{1}{4}}$
$y = ERROR$

and thus as vonflex1 said: $y^2$ can't be negative

4. there is an imaginary answer

$
y = \frac{\sqrt{-1}}{\sqrt{4}} = \frac{i}{2}
$

5. Ok,this simple equation is part of bigger complex numbers task:

and general solution for this task is:

than solution for must be in order to match general solution of this task?

6. not sure how that works?
$\left(-\frac{1}{2}\right)^2 \ne -\frac{1}{4}$

but

$\left(-\frac{i}{2}\right)^2 = -\frac{1}{4}$

7. Try to solve entire problem,than you will get to the part where

and I just don't know how solution of the task is ???

Ok,I agree about x value,but how did they get that y value? ???

Try to solve entire problem,than you will get to the part where

and I just don't know how solution of the task is ???

Ok,I agree about x value,but how did they get that y value? ???
Your original relation can be written as $|z| = |z + 1|$ and this defines a line (the perpendicular bisector of the line segment joining z = 0 and z = -1). If you substitute z = x + iy the same answer is easily got.

I have no idea where all your other stuff has come from since you refuse to show any of your working.

9. $|Z|=-\frac{1}{2}+ki$

$|Z+1|=\frac{1}{2}+ki$

$|Z|=\sqrt{(-0.5)^2+k^2}=\sqrt{\frac{1}{4}+k^2}$

$|Z+1|=\sqrt{(0.5)^2+k^2}=\sqrt{\frac{1}{4}+k^2}$