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- Astrophysics

Difference of squares help

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- Astrophysics

Difference of squares help

- jamiebookeater

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- Astrophysics

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 :)

- Astrophysics

@Empty Show me your squareception method!
@ganeshie8

- ganeshie8

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

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- Empty

Well specifically for this one haha: |dw:1436933157294:dw| 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:
\[(2b+a-b)(a-b)\]
:)

- ganeshie8

you don't want to use distributive law is it

- Astrophysics

Yeah pretty much, that's nice empty haha. I just want to see if there are other methods that even a 5 year old could understand

- Astrophysics

Actually it can be complicated as well, I just want to see what importance it holds :P

- jtvatsim

Here's another geometric derivation:
|dw:1436933365159:dw|