# Simplifying Expressions

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• Oct 22nd 2011, 07:10 AM
richtea9
Simplifying Expressions
Hi Folks,

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

$(x+1)^2 - (x-1)^2$
• Oct 22nd 2011, 07:14 AM
Quacky
Re: Simplifying Expressions
Are you aware of the difference of two squares, and how it applies here?
This states that:
$a^2-b^2=(a+b)(a-b)$
• Oct 22nd 2011, 07:17 AM
richtea9
Re: Simplifying Expressions
Quote:

Originally Posted by Quacky
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.
• Oct 22nd 2011, 07:19 AM
Prove It
Re: Simplifying Expressions
Quote:

Originally Posted by richtea9
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.
• Oct 22nd 2011, 07:19 AM
Quacky
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! (Bow)
• Oct 22nd 2011, 07:25 AM
richtea9
Re: Simplifying Expressions
Quote:

Originally Posted by Quacky
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! (Bow)

Ok, this is what I got using the foil method.

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

Not sure though if its correct.
• Oct 22nd 2011, 07:29 AM
Quacky
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!
• Oct 22nd 2011, 07:35 AM
richtea9
Re: Simplifying Expressions
Quote:

Originally Posted by Quacky
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. (Worried)

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

Thanks for the help so far btw.
• Oct 22nd 2011, 07:43 AM
Quacky
Re: Simplifying Expressions
That's fine (Happy).

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?
• Oct 22nd 2011, 07:53 AM
richtea9
Re: Simplifying Expressions
Quote:

Originally Posted by Quacky
That's fine (Happy).

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 $4x$
• Oct 22nd 2011, 08:08 AM
Quacky
Re: Simplifying Expressions
Quote:

Originally Posted by richtea9
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.

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

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.
• Oct 22nd 2011, 08:26 AM
richtea9
Re: Simplifying Expressions
Quote:

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

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

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. :confused:

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

Where have the brackets around the -2x come from?
• Oct 22nd 2011, 09:26 AM
Quacky
Re: Simplifying Expressions
Quote:

Originally Posted by richtea9
Thanks again,

However I'm still confused by all of it. :confused:

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 $-1(...)$
• Oct 22nd 2011, 09:41 AM
Prove It
Re: Simplifying Expressions
Quote:

Originally Posted by richtea9
Thanks again,

However I'm still confused by all of it. :confused:

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 $\displaystyle (x^2 - x^2) + [2x - (-2x)] + (1 - 1)$...
• Oct 22nd 2011, 11:19 AM
richtea9
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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