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That Cubic Formula

  1. Dec 17, 2009 #1

    Char. Limit

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    Gold Member

    The proof of the quadratic formula was so simple, I moved to the proof of the cubic formula with supreme confidence. And found myself awash in as and cs and cubic roots.

    Can you turn this equation into a cubic?

    [tex]x=-\frac{b}{3a}[/tex]

    [tex]-\frac{1}{3a}\sqrt[3]{\frac{1}{2}(2b^3-9abc+27(a^2)d+\sqrt{(2b^3-9abc+27(a^2)d)^2-4(b^2-3ac)^3}}[/tex]

    [tex]-\frac{1}{3a}\sqrt[3]{\frac{1}{2}(2b^3-9abc+27(a^2)d-\sqrt{(2b^3-9abc+27(a^2)d)^2-4(b^2-3ac)^3}}[/tex]
     
  2. jcsd
  3. Dec 18, 2009 #2
    Googling "proof of cubic forumla" gives this proof.
     
  4. Mar 25, 2010 #3
    First, the cubic equation: [tex]ax^3+bx^2+cx+d[/tex]. With [tex]x=y-\frac{a}{3}[/tex], you can reduce the equation to [tex]y^3+py+q[/tex]. [tex]p=b-\frac{a^2}{3}[/tex] and [tex]q=c-\frac{ab}{3}+\frac{2a^3}{27}[/tex]. In a cubic equation there are 3 possible answers, the one you listed would be one of the 3, [tex]X_{1}[/tex]
     
    Last edited: Mar 25, 2010
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