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Thread: Quadratics ( i think )

  1. #1
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    Quadratics ( i think )

    What is the minimum product of two numbers whose difference is 22? What numbers yield this product?


    FYI i asked my teacher and she didn't quite explain it the way i hoped so i was hoping someone could explain it better ?
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  2. #2
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    Hello, diehardmath4!

    What is the minimum product of two numbers whose difference is 22?
    What numbers yield this product?

    Let x = the larger number.
    Let y = the smaller number.

    Their difference is 22: . x - y \:=\:22 \quad\Rightarrow\quad  y \:=\:x-22 .[1]

    Their product is: . P \:=\:xy .[2]

    Substitute [1] into [2]: . P \;=\;x(x-22)\quad\Rightarrow\quad P \:=\:x^2 - 22x

    This is an up-opening parabola; its minimum is at its vertex.

    The vertex is at: . x \;=\;\frac{\text{-}b}{2a} \;=\;\frac{22}{2(1)} \quad\Rightarrow\quad \boxed{x\:=\:11}

    Substitute into [1]: . y \:=\:11-22\quad\Rightarrow\quad\boxed{ y \:=\:\text{-}11}


    The minimum product is: . (11)(\text{-}11) \;=\;\boxed{\text{-}121}

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  3. #3
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    thanks

    thank you
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