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Math Help - quad

  1. #1
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    quad

    find the real roots of
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  2. #2
    Senior Member I-Think's Avatar
    Joined
    Apr 2009
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    288

    Important idea

    An important idea when solving questions with infinite nested radicals.

    If x=\sqrt{a\sqrt{a\sqrt{a...}}}

    Then we can rewrite the equation as x=\sqrt{ax}
    As it is as infinite succession, we are allowed to 'cut' the expression anywhere convenient and let it be equal to the L.H.S.

    Example (from Pi in the Sky: Issue 11)

    x=\sqrt{3\sqrt{2\sqrt{3\sqrt{2...}}}}. Find x^3

    Answer
    x=\sqrt{3\sqrt{2x}}
    x^2=3\sqrt{2x}
    x^4=18x
    x^3=18

    With this technique, you could solve your question now, I'll give another example then move on to your question.

    x=\sqrt{2\sqrt{2\sqrt{2...}}}
    x^2=2x
    x=2
    In general, if x=\sqrt{a\sqrt{a\sqrt{a...}}}
    Then x^2=ax
    And x=a

    Now to your question
    \sqrt{x+2\sqrt{x+2\sqrt{x+...2\sqrt{x+2\sqrt{3x}}}  }}=x
    Then using our technique, we convert to
    \sqrt{x+2x}=x
    x^2=3x
    x^2-3x=0
    x(x-3)=0

    The real roots are 0 and 3.

    Don't hesitate to ask anymore questions.
    God bless your mathematical endeavours.
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