Solution 2.3:6b
From Förberedande kurs i matematik 1
By completing the square, the second degree polynomial can be rewritten as a quadratic plus a constant, and then it is relatively straightforward to read off the expression's minimum value,
\displaystyle x^{2}-4x+2=\left( x-2 \right)^{2}-2^{2}+2=\left( x-2 \right)^{2}-2
Because
\displaystyle \left( x-2 \right)^{2}
is a quadratic, this term is always larger than or equal to
\displaystyle 0
and the whole expression is therefore at least equal to
\displaystyle -\text{2}, which occurs when
\displaystyle x-\text{2}=0\text{ }
and the quadratic is zero, i.e.
\displaystyle x=\text{2}.
