From Förberedande kurs i matematik 2

When we complete the square of the second degree expression, \displaystyle z^2+2z+3, we start with the forumla

\displaystyle (z+a)^2 = z^2+2az+a^2

which we write as

\displaystyle (z+a)^2-a^2 = z^2+2az

and adapt the constant to be \displaystyle a=1 so that the terms \displaystyle z^2+2z are equal to \displaystyle z^2+2az, and therefore can be written as \displaystyle (z+1)^2-1^2. The whole calculation becomes

\displaystyle \underline{z^2+2z\vphantom{()}}+3 = \underline{(z+1)^2-1^2}+3 = (z+1)^2 + 2\,\textrm{.}

The underlined terms show the actual completion of the square.