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Math Help - Simplifying Expressions

  1. #1
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    Simplifying Expressions

    Hi Folks,

    Wondering you guys can show me and help me simplify this expression.

    (x+1)^2 - (x-1)^2
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  2. #2
    Super Member Quacky's Avatar
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    Re: Simplifying Expressions

    Are you aware of the difference of two squares, and how it applies here?
    This states that:
    a^2-b^2=(a+b)(a-b)
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  3. #3
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    Re: Simplifying Expressions

    Quote Originally Posted by Quacky View Post
    Are you aware of the difference of two squares, and how it applies here?
    This states that:
    a^2-b^2=(a+b)(a-b)
    I am not. I only learnt how to expand expressions yesterday with the foil method, so my knowledge is very basic.
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  4. #4
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    Re: Simplifying Expressions

    Quote Originally Posted by richtea9 View Post
    I am not. I only learnt how to expand expressions yesterday with the foil method, so my knowledge is very basic.
    In that case, take note that (a + b)^2 = (a + b)(a + b)

    Use FOIL to expand each of the squares, then collect like terms.
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  5. #5
    Super Member Quacky's Avatar
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    Re: Simplifying Expressions

    Ok. Well then let's not run before we can walk. What did you get when you tried to expand? We have:

    (x+1)(x+1)-(x-1)(x-1)

    Use the FOIL method twice.

    Edit: Prove It's elite typing skills squash me again!
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  6. #6
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    Re: Simplifying Expressions

    Quote Originally Posted by Quacky View Post
    Ok. Well then let's not run before we can walk. What did you get when you tried to expand? We have:

    (x+1)(x+1)-(x-1)(x-1)

    Use the FOIL method twice.

    Edit: Prove It's elite typing skills squash me again!
    Ok, this is what I got using the foil method.

    (x^2 + 2x +1) - (x^2 - 2x - 1)

    Not sure though if its correct.
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  7. #7
    Super Member Quacky's Avatar
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    Re: Simplifying Expressions

    Nearly. It should be:

    (x^2+2x+1)-(x^2-2x+1) because -1\times{-1}=1

    Can you see how to simplify further? You need to group like terms - but be wary of that negative sign!
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  8. #8
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    Re: Simplifying Expressions

    Quote Originally Posted by Quacky View Post
    Nearly. It should be:

    (x^2+2x+1)-(x^2-2x+1) because -1\times{-1}=1

    Can you see how to simplify further? You need to group like terms - but be wary of that negative sign!
    Sorry, I cannot see how to simplify further.

    Is it something to do with the x^2 + 2x and x^2 - 2x

    Thanks for the help so far btw.
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  9. #9
    Super Member Quacky's Avatar
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    Re: Simplifying Expressions

    That's fine .

    We have x^2+2x+1-(x^2-2x+1)

    I am going to start by getting rid of the brackets on the right by multiplying through by the negative. This will give:

    x^2+2x+1-x^2+2x-1

    Take a moment to make sure you digest that - look at each of the signs.

    As addition is commutative, we can rewrite this as:

    x^2-x^2+2x+2x+1-1

    Can you see anything here?
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  10. #10
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    Re: Simplifying Expressions

    Quote Originally Posted by Quacky View Post
    That's fine .

    We have x^2+2x+1-(x^2-2x+1)

    I am going to start by getting rid of the brackets on the right by multiplying through by the negative. This will give:

    x^2+2x+1-x^2+2x-1

    Take a moment to make sure you digest that - look at each of the signs.

    As addition is commutative, we can rewrite this as:

    x^2-x^2+2x+2x+1-1

    Can you see anything here?
    I'm unsure what has happened to the brackets on the first expression you provided and what do you meen by 'multiplying through by the negative'.

    Looking at the last expression you provided, this is how I see it, but it's properly completely wrong.

    x^2 - x^2 cancels each other out, so can we remove these?

    2x + 2x would that become 4x

    and finally, as 1 - 1 is = 0, we remove these? Leaving just 4x
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  11. #11
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    Re: Simplifying Expressions

    Quote Originally Posted by richtea9 View Post
    I'm unsure what has happened to the brackets on the first expression you provided and what do you meen by 'multiplying through by the negative'.

    Looking at the last expression you provided, this is how I see it, but it's properly completely wrong.

    x^2 - x^2 cancels each other out, so can we remove these?

    2x + 2x would that become 4x

    and finally, as 1 - 1 is = 0, we remove these? Leaving just 4x
    4x is right, and you can't simplify it any further.

    To clarify, we had:

    x^2+2x+1{\color{red}-}(x^2-2x+1)

    This means " x^2+2x+1 minus every individual term in the brackets"

    So we can distribute the - sign to every individual term in the brackets. It's exactly the same as multiplying through by -1, which is why I imprecisely said "multiply through by the negative". Basically, you just need to change all of the signs in the function one by one.

    =x^2+2x+1{\color{red}-}x^2{\color{red}-}(-2x){\color{red}-}+1

    =x^2+2x+1-x^2+2x-1

    Which I then rearranged to help simplify further.
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  12. #12
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    Re: Simplifying Expressions

    Quote Originally Posted by Quacky View Post
    4x is right, and you can't simplify it any further.

    To clarify, we had:

    x^2+2x+1{\color{red}-}(x^2-2x+1)

    This means " x^2+2x+1 minus every individual term in the brackets"

    So we can distribute the - sign to every individual term in the brackets. It's exactly the same as multiplying through by -1, which is why I imprecisely said "multiply through by the negative". Basically, you just need to change all of the signs in the function one by one.

    =x^2+2x+1{\color{red}-}x^2{\color{red}-}(-2x){\color{red}-}+1

    =x^2+2x+1-x^2+2x-1

    Which I then rearranged to help simplify further.
    Thanks again,

    However I'm still confused by all of it.

    I understand up until point of where you minus every individual term in the bracket.

    Where have the brackets around the -2x come from?
    Last edited by richtea9; October 22nd 2011 at 08:44 AM.
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  13. #13
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    Re: Simplifying Expressions

    Quote Originally Posted by richtea9 View Post
    Thanks again,

    However I'm still confused by all of it.

    I understand up until point of where you minus every individual term in the bracket.

    Where have the brackets around the -2x come from?
    Those brackets were unnecessary - I just put them there to clarify. --2x is ambiguous.

    I've been trying to think of another way to explain it without success. I think, honestly, you need someone else to provide a different perspective here.

    Consider -(6+4)
    We can either say that this is -(10) by doing the addition inside the brackets first
    Or we can say that this is -6-4=-10.

    Both are perfectly valid and get the correct solution. As long as you perform the same operation to everything within the bracket, your result will be the right one.

    Consider 2(x+4)
    This means ' 2 multiplied by everything in the bracket'.
    Therefore, we could write it as 2x+8

    Consider \frac{(3x+6)}{3}
    This is saying 'everything inside the bracket divided by 3 and could be rewritten as (x+2).

    -(x^2-2x+1) just means 'subtract everything inside the bracket.'
    So we -x^2
    we - the -2x
    and we -1
    But because the 2x is already negative, when we do the subtraction, it becomes positive.

    If you're still confused, try considering -2(3x+5)
    This can be rewritten as -2(3x)+(-2)5
    =-6x-10
    Can you follow that? Your example is extremely similar, except that we just have -1(...)
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  14. #14
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    Re: Simplifying Expressions

    Quote Originally Posted by richtea9 View Post
    Thanks again,

    However I'm still confused by all of it.

    I understand up until point of where you minus every individual term in the bracket.

    Where have the brackets around the -2x come from?
    \displaystyle (x^2 + 2x + 1) - (x^2 - 2x + 1)

    You need to subtract ALL of what is in the second set of brackets from what is in the first set of brackets...

    So you have \displaystyle (x^2 - x^2) + [2x - (-2x)] + (1 - 1)...
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  15. #15
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    Re: Simplifying Expressions

    Ok, I've been working on the following expression to see if I could get the right result.

    (x-2)^2+4x

    So I first do:

    (x-2)(x-2)
    = x^2-2x-2x-4+4x
    = x^2-4

    So I think the result is this? x^2-4

    If this is wrong, hopefully you should be able to see the thought process I have gone through and figure out what I'm doing wrong.
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