1. $[2]$ Find the minimum of $x^{2}-2 x$ over all real numbers $x$.
Answer: -1 Write $x^{2}-2 x=x^{2}-2 x+1-1=(x-1)^{2}-1$. Since $(x-1)^{2} \geq 0$, it is clear that the minimum is -1 .
Alternate method: The graph of $y=x^{2}-2 x$ is a parabola that opens up. Therefore, the minimum occurs at its vertex, which is at $\frac{-b}{2 a}=\frac{-(-2)}{2}=1$. But $1^{2}-2 \cdot 1=-1$, so the minimum is -1 .