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Math Help - Another quadratic expression

  1. #1
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    Another quadratic expression

    Prove that a^4+b^4+a^2b^2+x^2-xy+y^2+1>0
    Help would be appreciated!
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  2. #2
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    Re: Another quadratic expression

    Hey RuyHayabusa.

    Hint: Try completing the square for the pairs of variables (a,b) and (x,y).
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  3. #3
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    Re: Another quadratic expression

    Quote Originally Posted by RuyHayabusa View Post
    Prove that a^4+b^4+a^2b^2+x^2-xy+y^2+1>0
    Help would be appreciated!
    The statement is not true.

    \displaystyle \begin{align*} a^4 + b^4 + a^2b^2 + x^2 - x\,y + y^2 + 1 &= \left( a^2 \right)^2 + 2a^2b^2 + \left( b^2 \right)^2 + x^2 - 2\,x\,y + y^2 + 1 - a^2b^2 + x\,y \\ &= \left( a^2 + b^2 \right)^2 + \left( x^2 - y^2 \right)^2 + 1 - a^2b^2 + x\,y \end{align*}

    For this to be positive, we require \displaystyle \begin{align*} x\,y - a^2b^2 > 1 \end{align*}. There is no guarantee of this.
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