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Math Help - x^2 + 1 = 2^x; solved for x??

  1. #1
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    x^2 + 1 = 2^x; solved for x??

    hi there,

    i've come to an impasse on this one

    x^2 + 1 = 2^x

    would like to solve it for x and have tried
    manipulating logs and all sorts of things.

    by the way i know the answer is 1
    i just want to know if the equation
    can be stated in terms of x.

    thanks,
    poly
    Last edited by polypus; July 13th 2007 at 12:48 PM. Reason: left something out
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  2. #2
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    Quote Originally Posted by polypus View Post
    hi there,

    i've come to an impasse on this one

    x^2 + 1 = 2^x
    Define the function f(x)=2^x - x^2 - 1.

    Then using derivative you can show where it is increasing and decreasing that will imply that x=0,1 are the only solutions.

    It is striaghtforward but a little messy I leave the details to you.
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  3. #3
    MHF Contributor red_dog's Avatar
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    It seems that the graphs of the functions f(x)=x^2+1 and g(x)=2^x have three points of intersection.
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    MHF Contributor red_dog's Avatar
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  5. #5
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by polypus View Post
    hi there,

    i've come to an impasse on this one

    x^2 + 1 = 2^x

    would like to solve it for x and have tried
    manipulating logs and all sorts of things.

    by the way i know the answer is 1
    i just want to know if the equation
    can be stated in terms of x.

    thanks,
    poly
    To answer your question directly, I don't believe that an analytic solution to the problem exists. All you can do is estimate and guess solutions.

    -Dan
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    Senior Member DivideBy0's Avatar
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    Quote Originally Posted by topsquark View Post
    To answer your question directly, I don't believe that an analytic solution to the problem exists. All you can do is estimate and guess solutions.

    -Dan
    Do you believe that one day an analytic solution will exist?
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  7. #7
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by DivideBy0 View Post
    Do you believe that one day an analytic solution will exist?
    When I said "I don't believe" I meant that "to my present level of understanding it is not true," not that my belief system doesn't include such a solution.

    -Dan
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  8. #8
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    thanks

    wow,

    thanks for all the quick responses! i will
    be back to this forum in the future i'm sure.

    glad it was not just my still shaky abilities
    contributing to the lack of a solution.

    is there some theorem or theory which
    would say definitively whether there is
    or is not an analytic solution to my
    particular question?

    thanks again,
    poly
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  9. #9
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    Quote Originally Posted by DivideBy0 View Post
    Do you believe that one day an analytic solution will exist?
    You can create your own non-elementary functions to create solutions.

    So for example,
    ax+b=e^x for ab\not =0

    Has solutions via "Lambert W function", that is an analytic solution.

    Same here we can create the "Hacker Z function" of whatever you call it and solve:
    ax^2+bx+c = e^x
    Or something like that.

    But it will not be one of those standard "nice" functions.
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