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  1. #1
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    help

    Prove or Disprove that there exists a real number x such that x^2+x+1 is less than or equal to zero.
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  2. #2
    Senior Member topher0805's Avatar
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    Do you know the intermediate value theorem?
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  3. #3
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    yes
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  4. #4
    Senior Member topher0805's Avatar
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    Why not just use the quadratic equation?


    <br />
x = \frac {-b\pm \sqrt{b^2 - 4ac}}{2a}
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  5. #5
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    Yes that gives me a solution but then how do i prove that the equation works? Thats where i am now stuck.
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  6. #6
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    From the quadratic equation, you know there are no real roots, meaning the function is never 0 (either always negative, or always positive). With that, you can note that letting x=0 gives you a value of 1, so the function is always positive.

    Not quite the most rigorous proof and probably not what your professor is looking for, but it is valid.

    Edit: Also, I forgot to add that this works because the function is continuous over all real numbers.
    Last edited by xifentoozlerix; March 27th 2008 at 02:49 PM.
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  7. #7
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by natewalker205 View Post
    Yes that gives me a solution but then how do i prove that the equation works? Thats where i am now stuck.
    Well the quadratic formula tells you where you have a value of x such that
    x^2 + x + 1 = 0
    which is one of the criterion of your proof. So the existence of x proves that the function can be less than or equal to 0.

    Likely the best method is the intermediate value theorem, if you can do it that way.

    -Dan
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