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Math Help - Where does this problem fail?

  1. #1
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    Question Where does this problem fail?

    So, someone was showing me a math problem where he said he could get 1 = 2. Here's what he did:

    x = 1

    Thus
    x^2 = x^2

    So
    x^2 - x = x^2 - 1
    Right?

    Then, he factored:
    x(x -1) = (x - 1)(x + 1)

    Divided by (x - 1):
    [x(x-1)]/(x-1) = [(x-1)(x + 1)]/(x - 1)
    to cancel out (x - 1)

    Resulted in:
    x = x + 1
    From where he plugged in x = 1:
    1 = 1 + 1
    Therefore:
    1 = 2

    Please explain to me why this fails, because it seems like it does in so many places, but I want to know why exactly! For instance, dividing by (x - 1) would be the same as dividing by zero, which would fail, but he thinks he's pretty much found where math has a weakness. I need expert verification!
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  2. #2
    Super Member Bacterius's Avatar
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    There is a division per zero while simplifying, which invalidates all the following, sorry. You cannot divide per zero, because it would mean that 2 = 1, crumbling the whole theory of mathematics. Look at this :

    a = b

    a^2 = ab (multiply by a)

    2a^2 - 2ab = a^2 - ab (add a^2 - 2ab)

    2(a^2 - ab) = a^2 - ab (factorize on left side)

    2 = 1 (simplify by a^2 - ab)

    Which is impossible, because a^2 - ab = 0 (division per zero).
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  3. #3
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    Ray, you are awesome! Thanks!

    Quick edit: When you say (add a^2 - 2ab), do you mean 2a^2 - 2ab?
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  4. #4
    Super Member Bacterius's Avatar
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    Quote Originally Posted by audax View Post
    Ray, you are awesome! Thanks!

    Quick edit: When you say (add a^2 - 2ab), do you mean 2a^2 - 2ab?
    No, I mean a^2 - 2ab
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  5. #5
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    can't divide by 0

    note that x-1=1-1=0, so you can't divide by 0, as you have done.
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