Aus Online Mathematik Brückenkurs 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 \Bigl(x+\frac{a}{2}\Bigr)^{2} - \Bigl(\frac{a}{2}\Bigr)^{2}\,\textrm{.}

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

If we use this formula, we obtain

\displaystyle x^{2}+2x = \Bigl(x+\frac{2}{2}\Bigr)^{2} - \Bigl(\frac{2}{2}\Bigr)^{2} = (x+1)^{2}-1

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

\displaystyle x^{2}+2x-1 = (x+1)^{2}-1-1 = (x+1)^{2}-2\,\textrm{.}

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

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

and see that the relation really holds.