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Completing the Square with Coefficient on X^2

  1. Jul 8, 2009 #1
    1. The problem statement, all variables and given/known data

    Consider all possible surfaces that can be formed from the variations of

    \[PlusMinus]x^2\[PlusMinus]4y^2\[PlusMinus]z^2+2x+8y+6z==-6,

    that is, find and describe all the different quadric surfaces you can make from this equation by using different signs on the degree 2 terms. For each one, find the standard form of the equation.

    I am having problems completing the square when the coefficient is negative.

    3. The attempt at a solution

    For the case where the y-coefficient is negative:

    (x^2 + 2x + 1) + (-4y^2 + 8y + _ ) + (z^2 +6z + 9) = -6 + 1 + _ + 9

    dividing by -4 gives: (y^2 - 2y + _ )

    Adding (b/2a)^2: (y^2 - 2y + 1)

    Condensing: (y - 1)^2

    Now, does the coefficient 4 come back as positive or negative, as in
    4(y - 1)^2 or -4(y - 1)^2

    Also, when adding to the other side of the equation, does the -4 multiply to the added quantity?

    Thanks for any help
     
  2. jcsd
  3. Jul 8, 2009 #2

    Mark44

    Staff: Mentor

    (-4y2 + 8y )
    = -(4y2 - 8y + __) + something

    You're going to add a positive number inside the parentheses, but you have really added a negative number, so to keep the expression in balance, add the positive of that negative number.

    Make sense?
     
  4. Jul 8, 2009 #3

    Mark44

    Staff: Mentor

    What else you should do to make your life simpler is to bring out the coefficient on the 2nd degree term so that you have
    -4(y2 - 2y + __) + something

    Just keep track of what you really added to the overall expression.
     
  5. Jul 8, 2009 #4
    Okay, thanks for helpin me out
     
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