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

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In summary, when completing the square for an integration problem in calculus, adding parentheses to separate the -1 results in a change of signs due to the associative property. The correct way to start is by dividing the coefficient of x by 2, squaring it, and adding it to the expression. This results in a square inside the parentheses. Then, the constant term outside the parentheses is adjusted to maintain the original expression. Finally, the negative sign at the front is distributed to the completed square, resulting in a neater expression. This process can be applied to any quadratic expression, such as x^2 + px + q = 0.
  • #1
LearninDaMath
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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?
 
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  • #2
"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.
 
  • #3
-(x^2+4x-4)+7 ?
 
  • #4
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
 
  • #5
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.
 
  • #6
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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  • #7
Think I got it:

-(x+2)^2 + 7? :)
 

1. How do you simplify the expression [-x^2 -4x+4-1]?

The expression [-x^2 -4x+4-1] can be simplified by first combining like terms. In this case, -x^2 and x^2 are like terms and can be combined to give 0. Similarly, -4x and 4x can be combined to give 0. Therefore, the expression simplifies to [0+0+4-1] which further simplifies to [3].

2. Why is the expression [-x^2 -4x+4-1] equivalent to [(x^2+4x-4)-1]?

The two expressions are equivalent because the terms -x^2 and x^2 are additive inverses of each other and therefore cancel out. Similarly, the terms -4x and 4x are additive inverses and cancel out. This leaves us with [(0+0+4)-1] which simplifies to [(4)-1] or [3].

3. What is the process for combining like terms in algebra?

To combine like terms, you simply add or subtract the coefficients of the terms while keeping the variable and exponent the same. For example, in the expression 3x^2+2x^2, the like terms are 3x^2 and 2x^2. Combining these terms gives us 5x^2. Similarly, in the expression 4x+5x, the like terms are 4x and 5x. Combining these terms gives us 9x.

4. Can you rearrange terms in an algebraic expression?

Yes, you can rearrange terms in an algebraic expression as long as you use the commutative and associative properties of addition. This means that you can change the order of the terms or group them in any way you want without changing the value of the expression. However, you cannot change the sign or the exponent of a term.

5. What are the rules for simplifying algebraic expressions?

The rules for simplifying algebraic expressions include combining like terms, using the distributive property, and applying the rules of exponents. Additionally, you should always follow the order of operations, which is PEMDAS (parentheses, exponents, multiplication and division from left to right, and addition and subtraction from left to right).

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