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What is auxiliary equation?

  1. Dec 16, 2011 #1
    Can anyone can give one example how cubic become quadratic?
    Or can you recommand me some books? (I am an university student.)
  2. jcsd
  3. Dec 16, 2011 #2

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    Hi athrun200! :smile:

    Here's a method where a cubic equation is reduced to a quadratic equation.

    Starting with [itex]x^3+3x+6=0[/itex].

    Substitute x=y+z, meaning you have a free choice for either y or z.
    So [itex](y+z)^3+3(y+z)+6=0[/itex]

    [itex]\Rightarrow (y^3+z^3+3y^2z+3yz^2) + 3(y+z)+6=0[/itex]

    [itex]\Rightarrow y^3+z^3+3(yz+1)(y+z)+6=0[/itex]

    Choose z such that [itex]yz+1=0[/itex], or [itex]z=-{1 \over y}[/itex]
    Then: [itex]y^3 - {1 \over y^3} + 6 = 0[/itex]
    [itex]\Rightarrow (y^3)^2 + 6(y^3) - 1 = 0[/itex]

    Solve as a quadratic equation and back substituting z gives:
    [tex]x=y+z=\sqrt[3]{-3 + \sqrt{10}} - {1 \over \sqrt[3]{-3 + \sqrt{10}}}[/tex]
    [tex]x=y+z=-\sqrt[3]{3 + \sqrt{10}} + {1 \over \sqrt[3]{3 + \sqrt{10}}}[/tex]

    Note that both solutions are the same root.
  4. Dec 16, 2011 #3
    Thx very much!
    which topic does it belong to? Calculus? or Algebra?
    Are there any textbook about this?
  5. Dec 16, 2011 #4

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    Hmm, not sure.
    *looking up calculus*
    No, not calculus.

    *looking up algebra*
    Yes, I think it's algebra!

    TBH, I learned it from an old book of my father, that he had while studying.
    I don't recall what it was called, but it was a thick volume with a purple cover and many yellowed pages... I loved it! :smile:
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