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

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Discussion Overview

The discussion revolves around the process of completing the square for the expression \(-x^2 - 4x + 3\) and the implications of manipulating signs and parentheses in algebraic expressions. Participants explore the transformations involved and clarify their understanding of the associative property in relation to sign changes.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant describes the process of completing the square and questions how adding parentheses affects the signs in the expression.
  • Another participant agrees with the initial assertion about the associative property but points out a missing minus sign in the expression.
  • Some participants express confusion about whether \(x^2 + 4x - 4\) can be considered a square and suggest starting with \(-x^2 - 4x + 3\) instead.
  • Further contributions explore the steps involved in completing the square, including dividing coefficients and squaring them, while questioning how to correctly account for the negative sign.
  • A later reply suggests a potential final form of the expression as \(-(x+2)^2 + 7\), indicating progress in understanding.

Areas of Agreement / Disagreement

Participants express varying levels of confusion and differing interpretations of the algebraic manipulations involved. There is no consensus on the correct approach or final expression, as multiple viewpoints and methods are presented.

Contextual Notes

Some participants note that the expression \(x^2 + 4x - 4\) is not a perfect square, which may affect the validity of certain steps in the completion process. The discussion also highlights the importance of correctly managing signs when factoring out negatives.

Who May Find This Useful

Students reviewing algebraic manipulation, particularly those focused on completing the square and understanding the implications of sign changes in expressions.

LearninDaMath
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If I want to complete the square with

(-x^{2}-4x+3) I would write

(-x^{2}-4x+(...) +3 - (...)) = (-x^{2}-4x+4+3-4) = (-x^{2}-4x+4-1) = (x^{2}+4x-4) - 1Why 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?
 
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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.
 
-(x^2+4x-4)+7 ?
 
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
 
verty said:
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

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.
 
NextElement said:
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.

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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Think I got it:

-(x+2)^2 + 7? :)
 

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