Motivation for $y(x - y) = \frac{1}{2}(x^2 - y^2) - \frac{1}{2}(x-y)^2$ It's a really simple identity and according to the text, I am reading, this is an obvious identity
$$y(x - y) = \frac{1}{2}(x^2 - y^2) - \frac{1}{2}(x-y)^2$$
Without expanding the RHS, I don't know how they came up with this.

Factor the first term on the right hand side: $\frac{1}{2} (x^2-y^2) = \frac{1}{2} (x+y)(x-y).$ Factor the second term: $(x-y)^2=(x-y)(x-y)$ since squaring a number is defined to be the product of that number with itself.
Then factor out $(x-y)$ from $\frac{1}{2} (x+y)(x-y)-\frac{1}{2}(x-y)(x-y)$ since it is a common term (remember to think of parenthetical expressions like $(x-y)$ as SINGLE terms). You should get the left hand side from there. 
