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Algebra help please

  1. Aug 22, 2009 #1
    Hi Guys,

    My algebra is a little rusty, and I have become stuck with this problem, would anyone mind showing me the right path..

    The problem is..

    a = SQRT(b/x)-cx

    And I need to solve for x

    Thanks in advance..

    Steven
     
  2. jcsd
  3. Aug 22, 2009 #2
    Adding cx to both sides and squaring gives a^2 +2acx + c^2*x^2 = b/x. Multiplying by x and subtracting by b gives c^2*x^3 - 2acx^2 +a^2*x - b = 0. Now you just have to find a root for this third degree polynomial.
     
  4. Aug 22, 2009 #3
    That's the hard part =(

    Steven
     
  5. Aug 22, 2009 #4
    For a polynomial ax^3 + bx^2 + cx + d = 0, the roots may be given by these:
    http://www.josechu.com/ecuaciones_polinomicas/cubica_solucion.htm
    It might be simpler to use Newton's method to come up with an expression that is approximately close. Even still, there is most likely a simpler way...I just can't see it right off the back.
     
    Last edited: Aug 22, 2009
  6. Aug 27, 2009 #5
    shouldn't that be:

    + 2acx^2
     
  7. Aug 27, 2009 #6


    Hey I never knew that there was an exact expression for the roots of a cubic polynomial . I infact never thought of a method apart from trying the factor theorem ( or using numerical technique) .

    How was it show that this is the solution ( I mean apart from back substitution) ?
    I thought it was in fact impossible to find the roots for a general polynomial equation of order > 2.
     
  8. Aug 27, 2009 #7
    Solution formulae for polynomials of up to fourth order were known by mid-sixteenth century. I believe Gerolamo Cardano compiled them all in his Ars Magna of 1545. See

    http://www-math.cudenver.edu/~wcherowi/courses/m4010/polynom2.pdf [Broken]

    for some background of how this all came about.

    Solution formulae for higher order polynomials were sought thereafter, until things came to a head in the early 19th century. In particular Paulo Ruffini, Niels Abel, and Evariste Galois separatelly discovered (and proved to varying degrees, Galois being the deepest) that no general solution formulae exist for polynomials of order 5 or higher. This is not say one cannot solve any particular such polynomial (and many can be) but rather that there does not exist a formulae that works on all polynomials of a certain degree (> 4).

    The theorem in particular that states the impossibility of finding such a formula is called the Abel-Ruffini Theorem. Ruffini's first attempt at the proof in 1799 was slightly flawed, but Abel fixed it not long after. His proof suffered from exploring unknown territory and was consequently clunky and arcane. Galois developed a rich and powerful body of theory to clearly attack this particular puzzle. Galois' work has lead to the development of the body of theory named after him (he died at the age of 20 in 1832 from a gunshot wound suffered in a pistol duel).

    Wolfram's Mathworld Links

    Cubic Formula

    http://mathworld.wolfram.com/CubicFormula.html

    Quartic Equation

    http://mathworld.wolfram.com/QuarticEquation.html


    --Elucidus
     
    Last edited by a moderator: May 4, 2017
  9. Sep 15, 2009 #8
    How about using a conjugate?????????????? I once saw a poster for the roots of a 5th (quintic) degree. That guy had a lot of extra time.
     
  10. Sep 16, 2009 #9
    Thank You very much Elucidus for the PDF link ... the steps involved seem very simple and clear (for the cubic equation ) ... but who would have thought to do those steps in that way ... not me anyway !
     
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