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Math Help - problem solving - square roots

  1. #1
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    problem solving - square roots

    This problem was in my calculus exercise, yet I don't understand why:

    Solve the equation :
    x=\sqrt{5+\sqrt{5+\sqrt{5+x}}}
    (Show there are not other solutions to the ones you found)

    Now, solving it regularly using simple algebra got me into an eighth-degree polynomial, which reminded my my teachers have just told us there are proofs for why there are no 'formulas' for solving polynomial equations higher than 5.

    Anyway, how can I solve this problem using calculus tools? I thought of doing the following thing:
    We know that x=\sqrt{5+\sqrt{5+\sqrt{5+x}}}, therefore:
    x=\sqrt{5+\sqrt{5+\sqrt{5+x}}}=\sqrt{5+\sqrt{5+\sq  rt{5+\sqrt{5+\sqrt{5+\sqrt{5+x}}}}}}.

    Let the first expression be a_3, ane the second expression be a_6. Therefore:

    x=a_{3n}=\sqrt{5+\sqrt{5+\sqrt{5+...}}}
    Therefore,
    lim(x)=lim(a_{3n})

    but how do I find it ?
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  2. #2
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    Does anyone have any idea ?

    This looks like a nice one...
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  3. #3
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    Well, if you think about it you can write:

    x = \sqrt{5+x} (can you understand why?)
    So all you need to do is solve the quadratic equation you get after squaring both sides.
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  4. #4
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    Quote Originally Posted by Defunkt View Post
    Well, if you think about it you can write:

    x = \sqrt{5+x} (can you understand why?)
    So all you need to do is solve the quadratic equation you get after squaring both sides.
    Hmm.. I can only write x = \sqrt{5+x}, then try to see if it works and get back again x = \sqrt{5+x}. I don't think this is how you got to it, and how I should understand it.

    This means it is a solution, but how can I know it's the only one?
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  5. #5
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    x = \sqrt{5+\sqrt{5+\sqrt{5+x}}} \Rightarrow x = \sqrt{5+\sqrt{5+\sqrt{5+\sqrt{5+\sqrt{5+\sqrt{5+x}  }}}}}<br />
\Rightarrow x = \sqrt{5+\sqrt{5+\sqrt{5+\sqrt{5+\sqrt{5+\sqrt{5+ \sqrt{...}}}}}}}

    So essentially, this is equivalent to x= \sqrt{5+x}, thus the solutions of the equation are the only ones which satisfy the condition for x.
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  6. #6
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    hmm, okay, that makes sense, but how can I be sure there is no other solution?
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