Algebra: How does [-x^2 -4x+4-1] become [(x^2+4x-4)-1]

  1. If I want to complete the square with

    [itex](-x^{2}-4x+3)[/itex] I would write

    [itex](-x^{2}-4x+(...) +3 - (...)) = (-x^{2}-4x+4+3-4) = (-x^{2}-4x+4-1) = (x^{2}+4x-4) - 1[/itex]


    Why does adding the parentheses to separate the -1 change all the signs. I understand it has something to do with factoring out a negative, but how exactly?

    I thought adding parentheses to a series of additions and/or subtractions is simply an associative property. Signs don't need to change in associate property, so why would they change when adding parentheses to separate the -1 when completing the square for an integration problem in calculus?
     
    Last edited: Aug 31, 2013
  2. jcsd
  3. arildno

    arildno 12,015
    Science Advisor
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    "I thought adding parentheses to a series of additions and/or subtractions is simply an associative property. Signs don't need to change in associate property, so why would they change when adding parentheses to separate the -1 when completing the square for an integration problem in calculus? "

    It doesn't; you are perfectly correct concerning addition/subtraction relative to the associative property.

    The last expression is missing a minus sign in front of the parenthesis expression containing the completed square.
     
  4. -(x^2+4x-4)+7 ?
     
  5. verty

    verty 1,897
    Homework Helper

    There's a mistake, x^2 + 4x - 4 is not a square.

    The correct way to start is:

    -x^2 - 4x + 3
    -(x^2 + 4x) + 3
     
  6. I'm confused as heck, but this is good practice since this is exactly what we're reviewing in math right now.

    If you start with -(x^2+4x) + 3, you divide that 4 by two and square it, resulting in -(x^2+4x+4)+3.

    However, you have do add that 4 to the outside, but doesn't the negative in the very front make it a negative 4, finally resulting in -(x^2+4x+4) + 7? I'm confused on where to go from here.
     
  7. verty

    verty 1,897
    Homework Helper

    You have done the hard work, you just need to write it in the neatest way possible. Remember you want to have something like (x+a)^2.

    Here is a more abstract example for you to practice the steps on:

    x^2 + px + q = 0
     
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  8. Think I got it:

    -(x+2)^2 + 7? :)
     
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