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Completing the square 2 variables

  1. Mar 17, 2012 #1
    1. The problem statement, all variables and given/known data
    Ax[itex]^{2}[/itex]+By[itex]^{2}[/itex]-Mxy

    Capitals are constants.

    The equation (one like it) is in a book I am reading where the author procedes to complete the square. He does it in one line without showing the procedure and I am completely baffled.

    2. Relevant equations



    3. The attempt at a solution
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Mar 17, 2012 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    Well, what exactly is the problem? Do you know how to complete the square?
    I will presume that you at least know that [itex](u+ v)^2= u^2+ 2uv+ v^2[/itex].
    Since you have [itex]Ax^2[/itex] and [itex]By^2[/itex] as [itex]u^2[/itex] and [itex]v^2[/itex], you must have [itex]u= \sqrt{A}x[/itex] and [itex]v= \sqrt{B}y[/itex] so that [itex]2uv= 2\sqrt{AB}xy[/itex].

    That is, we can write
    [tex]Ax^2+B^2- Mxy= Ax^2+ 2\sqrt{AB}xy+ By^2- Mxy- 2\sqrt{AB}xy[/tex]
    [tex]= (\sqrt{A}x+ \sqrt{B}y)^2- (Mxy+ 2\sqrt{AB}xy)[/tex]
     
  4. Mar 17, 2012 #3
    Here it is. I am pleased to say it all makes sense now.
     

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