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Astrophysics

  • one year ago

Difference of squares help

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  1. Astrophysics
    • one year ago
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    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 :)

  2. Astrophysics
    • one year ago
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    @Empty Show me your squareception method! @ganeshie8

  3. ganeshie8
    • one year ago
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    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

  4. Empty
    • one year ago
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    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: \[(2b+a-b)(a-b)\] :)

  5. ganeshie8
    • one year ago
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    you don't want to use distributive law is it

  6. Astrophysics
    • one year ago
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    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

  7. Astrophysics
    • one year ago
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    Actually it can be complicated as well, I just want to see what importance it holds :P

  8. jtvatsim
    • one year ago
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    Here's another geometric derivation: |dw:1436933365159:dw|