From Förberedande kurs i matematik 1

If we consider the rule

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

and move \displaystyle a^{2} over to the left-hand side, we obtain the formula

\displaystyle (x-a)^{2}-a^{2} = x^{2}-2ax\,\textrm{.}

With the help of this formula, we can rewrite (complete the square of) a mixed expression \displaystyle x^{2}-2ax to a obtain a quadratic expression, \displaystyle (x-a)^{2}-a^{2}.

The expression \displaystyle x^{2}-2x corresponds to \displaystyle a=1 in the formula above and thus

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