Is this a correct derivation of completing the square? $x^2 + bx$
$=x^2 + bx + c - c$
$=(x + k)^2 - c$
$=x^2 + 2kx + (k^2 - c) = x^2 + bx + 0$
This implies:
$2k = b$, so $k = b/2$, and:
$k^2 - c = 0$, or $k^2 = c$, or $(b/2)^2 = c$
So to complete the square we are making the transformation:
$x^2 + bx \implies (x + b/2)^2 - (b/2)^2$

Okay.  You are trying to derive.
As you want to derive $x^2 + bx = (x + k)^2 + c$ I'd start with that line:
"we want $x^2 + bx = (x+k)^2 +c$ and we need to find the $c$ and $k$ that will make it so.
You started instead with
$x^2 + bx = x^2 + bx + c - c = (x+k)^2 -c$ with, I guess (????) then assumption that $(x+k)^2 = x^2 + bx + c$.  But without any reason I don't get what you are doing.  What is that $c$.  It just came out of nowhere.  Why do you think that $x + bx + c = (x + k)^2$ where did the $k$ come from?  What are you doing? 
It doesnt seem clear.
As you want to solve $x^2 + bx = (x + k)^2 -c$ then start there.  It's almost as though you are afraid to ask the right question so you ask the wrong question $x^2 + bx + c -c = (x + k)^2 - c$.
But if you say:
"$x^2 + bx = (x+k)^2 -c $; what are the $k$ and $c$ we need to make the happen?"
The solution is clear.
$(x+k)^2 - c = x^2 + 2kx + k^2 -c$ and we want $x^2 + bx = x^2 + 2kx +k^2 - c$ so $k = \frac b2$ and $k^2 - c = \frac {b^2}4 - c = 0$.  So $c = -\frac {b^2}4$.
