<s>[INST] 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:
$x^2 + bx \implies (x + b/2)^2 - (b/2)^2$
 [/INST] It's true, but you can get it much  more easier:
$$x^2+bx=x^2+bx+\frac{b^2}{4}-\frac{b^2}{4}=\left(x+\frac{b}{2}\right)^2-\left(\frac{b}{2}\right)^2$$
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