Solution 3.3:3a
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 squaring rule
\displaystyle \left( z+a \right)^{2}=z^{2}+2az+a^{2}
which we write as
\displaystyle \left( z+a \right)^{2}-a^{2}=z^{2}+2az
and adapt the constant to be
\displaystyle a=\text{1 }
so that the terms
\displaystyle z^{2}+2z
are equal to
\displaystyle z^{2}+2az,, and therefore can be written as
\displaystyle \left( z+1 \right)^{2}-1^{2}. The whole calculation becomes
\displaystyle \underline{z^{2}+2z}+3=\underline{\left( z+1 \right)^{2}-1^{2}}+3=\left( z+1 \right)^{2}+2
The underline terms show the actual completion of the square.
