From Förberedande kurs i matematik 1

When we complete the square, it is only the first two terms, \displaystyle x^{2}+2x , that are involved. The general formula for completing the square states that \displaystyle x^{2}+ax equals

\displaystyle \left( x+\frac{a}{2} \right)^{2}-\left( \frac{a}{2} \right)^{2}

Note how the coefficient \displaystyle a in front of the \displaystyle x turns up halved in two places.

If we use this formula, we obtain

\displaystyle x^{2}+2x=\left( x+\frac{2}{2} \right)^{2}-\left( \frac{2}{2} \right)^{2}=\left( x+1 \right)^{2}-1

and if we subtract the last " \displaystyle 1 " , we obtain

\displaystyle x^{2}+2x-1=\left( x+1 \right)^{2}-1-1=\left( x+1 \right)^{2}-2

To be completely certain that we have used the correct formula, we can expand the quadratic on the right-hand side,

\displaystyle \left( x+1 \right)^{2}-2=x^{2}+2x+1-2=x^{2}+2x-1

and see that the relation really holds.