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  1. #1
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    algebra problem

    How can we show that there are no (x,y) pairs of real numbers that satisfy xy = 3 and (x + y)^2 = 10 ? Intuitively it's obvious but I'm not sure exactly how to show it..
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  2. #2
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    Quote Originally Posted by sarahh View Post
    How can we show that there are no (x,y) pairs of real numbers that satisfy xy = 3 and (x + y)^2 = 10 ? Intuitively it's obvious but I'm not sure exactly how to show it..
    Brute force:

    Sub y = 3/x into (x + y)^2 = 10.

    After expanding and simplifying you get x^2 + \frac{9}{x^2} = 4. Multiply through by x^2 and re-arrange:

    x^4 - 4x^2 + 9 = 0

    \Rightarrow (x^2 - 2)^2 - 4 + 9 = 0

    \Rightarrow (x^2 - 2)^2 = -5.

    Therefore no real solutions.
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