From Förberedande kurs i matematik 1

Using the squaring rule, we recognize the polynomial as the expansion of \displaystyle \left( x-1 \right)^{2},

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

This quadratic expression has its smallest value, zero, when \displaystyle x-\text{1}=0, i.e. \displaystyle x=\text{1}. All non-zero values of \displaystyle x-\text{1} give a positive value for \displaystyle \left( x-1 \right)^{2}.

NOTE: If we draw the curve \displaystyle y=\left( x-1 \right)^{2}, we see that it has a minimum value of zero at \displaystyle x=\text{1}.