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Math Help - Prove equation is -ve for all real values

  1. #1
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    Prove equation is -ve for all real values

    Hi Guys,

    This problem from quadratic equations exercises that I am doing. Not sure how to proceed. Your help is much appreciated.

    Prove that the expression (24 - x)(x - 8) - k is negative for all real values of x provided that k is greater than 64. Hence show that the expression (6 + y)(4 - y)(y + 4)(y - 2) - 65 is negative for all real values of y.

    The first part I solved by completing the square to get -(x + 16)^2 + (64 - k). So -ve for k > 64.

    Having trouble with the rest of the question. I tried simplifying but I end with an equation for order 4. What am I doing wrong?

    Thanks for your help!
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  2. #2
    MHF Contributor red_dog's Avatar
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    (6+y)(4-y)(4+y)(y-2)-65=(24-y^2-2y)(y^2+2y-8)-65

    Noe substitute y^2+2y with x and you get (24-x)(x-8)-65, then apply the first part of the problem.
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  3. #3
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    Quote Originally Posted by red_dog View Post
    (6+y)(4-y)(4+y)(y-2)-65=(24-y^2-2y)(y^2+2y-8)-65

    Noe substitute y^2+2y with x and you get (24-x)(x-8)-65, then apply the first part of the problem.
    Thanks @red_dog, seems so simple now!
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