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Completing the square

  1. Dec 14, 2011 #1
    How do I complete the square for: [tex]x^2 + 2y^2 + 2z^2 + 4xy +4xz + 4yz[/tex]
    I can do it for a normal quadratic [itex]ax^2 + bx + c[/itex] but how do I do it for something like this with more than one variable?
     
  2. jcsd
  3. Dec 14, 2011 #2

    SammyS

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    I doubt that there is one particular "correct" result for this.

    Notice that this polynomial does contain a perfect square for two of the variables, y & z: [itex]2y^2+4yz+2z^2=2(y+z)^2\,. [/itex]

    Factor 4x out of the two remaining "mixed" terms.
    [itex]x^2+4x(y+z)+2(y+z)^2[/itex]​
    I see two ways to finish. You can figure out how many terms of (y+z)2 to add & subtract, or you can figure out how many terms of x2 to add & subtract. The latter way gives a simpler looking answer.
     
  4. Dec 14, 2011 #3
    I've got it equal to the sum of squares: [tex]2(x+y+z)^2 - x^2[/tex] What does this mean the rank and signature is? There is 1 positive square and 1 negative square so is the rank 1+1=2 and the signature 1-1=0?
     
  5. Dec 14, 2011 #4

    Ray Vickson

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    You can apply (essentially) the algorithm that leads to Cholesky factorization of a matrix. In your example, start by looking at the x-terms, which are [itex] x^2 + 4xz + 4yz .[/itex] These are what you would get if you looked at [itex] (x+2y+2z)^2, [/itex] because that is where you would get terms of the form 4xy and 4xz. Anyway, we have [tex] x^2 + 4xy + 4xz = (x+2y + 2z)^2 - 4y^2 - 4z^2 - 8yz, [/tex], so your function f(x,y,z) is
    [tex] (x+2y+2z)^2 -2y^2 - 2z^2 -4yz. [/tex] Now [tex]2y^2+2z^2+4yz = 2(y+z)^2,[/tex]
    so we finally have [tex] f(x,y,z) = (x+2y+2z)^2-2(y+z)^2. [/tex]

    RGV
     
  6. Dec 14, 2011 #5
    So is the rank 2 and the signature 0?
     
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