Question

Difference of squares help

Answer 1

Hey, so I've been wondering about this for a while...we know that \[a^2-b^2 = (a+b)(a-b)\] but that's through distributing, so I have a question, say we never knew the existence of \[(a+b)(a-b)\] how exactly would we prove/ derive \[a^2-b^2 = (a+b)(a-b)\] I know there maybe geometry but is there a way using algebra, other methods, actually anything will be useful, as I don't find anything intuitive about \[a^2-b^2\] and I don't see exactly how we can derive it mathematically/ or what exactly it would mean without the whole \[(a+b)(a-b)\]

Thanks :)

Answer 2

@Empty Show me your squareception method!

@ganeshie8

Answer 3

More generally : \[x^n-y^n = (x-y)(x^{n-1}+x^{n-2}y+\cdots+xy^{n-2}+y^{n-1})\]

but this might be a sledgehammer for the original problem

Answer 4

Well specifically for this one haha: So the place we want is those two rectangles and the little square there:

\[2b(a-b) +(a-b)^2\]
factor out an (a-b) term:
\[(2b+a-b)(a-b)\]
:)