$x$ is an element of $\mathbb{Z}$, for all values of $x$, $x^2 - x \geq 0$

Question

I understand why this works, the proof does not make sense to me though:

How does he get the $-\dfrac14$, and why does it prove that the statement is true?

Answer 1

If you want a more hands on approach, let's try cases. Note $x^2-x=x(x-1)$. Now if $x\geq 2$, both $x$ and $x-1$ are positive, so their product is as well. What happens if $x<0$? What about the remaining two cases?