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Impossible to solve for x. What to do now?

  1. Feb 7, 2010 #1
    1. The problem statement, all variables and given/known data

    Solve this equation:

    [tex]x^2 = 2 \sqrt{x^3 + 1}[/tex]


    3. The attempt at a solution

    Squaring both sides:

    [tex]( x^2 )^2 = ( 2 \sqrt{x^3 + 1} )^2[/tex]

    [tex]x^4 = 4( x^3 + 1 )[/tex]

    [tex]x^4 = 4x^3 + 4[/tex]

    [tex]x^4 - 4x^3 = 4[/tex]

    Now what? Is there is any technique in whole of mathematics with which we can find an apporiximate solution, if not the actual value?
     
  2. jcsd
  3. Feb 7, 2010 #2

    radou

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  4. Feb 7, 2010 #3

    vela

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    You could use Newton-Raphson to find the roots. If you have an approximation [itex]x_n[/itex] for the root, you can get a new approximation by calculating

    [tex]x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}[/tex]

    When it works, it converges on the root quickly. You can get initial guesses by plotting the function.
     
  5. Feb 7, 2010 #4
    Is Newton-Raphson method the best method available for approximation?
     
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