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Math Help - Proving

  1. #1
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    Proving

    How do you prove that x^2+y^2-2xy+x-y+1 is greater than 0?
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  2. #2
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    Re: Proving

    You can't.

    \displaystyle \begin{align*} x^2 - 2\,x\,y + y^2 + x - y + 1 &= \left( x - y \right)^2 + x - y + 1 \end{align*}

    All that you can say for sure is that \displaystyle \begin{align*} \left( x - y \right)^2 \end{align*} is nonnegative. The entire quantity will only be positive if \displaystyle \begin{align*} x - y + 1 > 0 \end{align*} , which you are not guaranteed.
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  3. #3
    Junior Member Nehushtan's Avatar
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    Re: Proving

    Quote Originally Posted by Prove It View Post
    You can't.

    \displaystyle \begin{align*} x^2 - 2\,x\,y + y^2 + x - y + 1 &= \left( x - y \right)^2 + x - y + 1 \end{align*}

    All that you can say for sure is that \displaystyle \begin{align*} \left( x - y \right)^2 \end{align*} is nonnegative. The entire quantity will only be positive if \displaystyle \begin{align*} x - y + 1 > 0 \end{align*} , which you are not guaranteed.
    It does not matter if x-y+1 is negative; (x-y)^2 may be big enough to make the whole expression positive.

    Indeed, x^2+y^2-2xy+x-y+1=(x-y)^2+(x-y)+1=\left(x-y+\frac12\right)^2+\frac34 is clearly always positive.
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  4. #4
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    Re: Proving

    Excellent proof ...My congratulations to Nehushtan

    Minoas
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