In going through old Monterey Physics Flash Games, I approached this current step in some derivation in connection to an elastic collision, where an identity was referenced.
However, I have never even heard of this identity; I can not find it online, and in plugging in $x=3$ and $y=5$ (arbitrarily) results in different values. Is this even a real identity? Why is it used if it is not true? If there is some truth in it, how would I prove it? I am in general confused as to why this constitutes a logical step in derivation if it is not even true. (As another point of consideration, is this identity true only for certain values?)
You should know that for any numbers
$$a\times (b+c)=a\times b+a\times c$$
Let’s apply that with $a=x-y$ and $b=x$, $c=y$
We then have
$$(x-y)\times (x+y)=(x-y)\times x+(x-y)\times y$$
Doing the same with $(x-y)\times x=x^2-y\times x$ and $(x-y)\times y=x\times y-y^2$ and adding all that together we get
$$(x-y)\times (x+y)=x^2+x\times y-y\times x-y^2=x^2-y^2$$