Hi Folks,
Wondering you guys can show me and help me simplify this expression.
$\displaystyle (x+1)^2 - (x-1)^2$
That's fine .
We have $\displaystyle x^2+2x+1-(x^2-2x+1)$
I am going to start by getting rid of the brackets on the right by multiplying through by the negative. This will give:
$\displaystyle x^2+2x+1-x^2+2x-1$
Take a moment to make sure you digest that - look at each of the signs.
As addition is commutative, we can rewrite this as:
$\displaystyle x^2-x^2+2x+2x+1-1$
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.
$\displaystyle x^2 - x^2$ cancels each other out, so can we remove these?
$\displaystyle 2x + 2x$ would that become $\displaystyle 4x$
and finally, as 1 - 1 is = 0, we remove these? Leaving just $\displaystyle 4x$
$\displaystyle 4x$ is right, and you can't simplify it any further.
To clarify, we had:
$\displaystyle x^2+2x+1{\color{red}-}(x^2-2x+1)$
This means "$\displaystyle x^2+2x+1$ minus every individual term in the brackets"
So we can distribute the $\displaystyle -$ sign to every individual term in the brackets. It's exactly the same as multiplying through by $\displaystyle -1$, 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.
$\displaystyle =x^2+2x+1{\color{red}-}x^2{\color{red}-}(-2x){\color{red}-}+1$
$\displaystyle =x^2+2x+1-x^2+2x-1$
Which I then rearranged to help simplify further.
Those brackets were unnecessary - I just put them there to clarify. $\displaystyle --2x$ 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 $\displaystyle -(6+4)$
We can either say that this is $\displaystyle -(10)$ by doing the addition inside the brackets first
Or we can say that this is $\displaystyle -6-4=-10$.
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 $\displaystyle 2(x+4)$
This means '$\displaystyle 2$ multiplied by everything in the bracket'.
Therefore, we could write it as $\displaystyle 2x+8$
Consider $\displaystyle \frac{(3x+6)}{3}$
This is saying 'everything inside the bracket divided by $\displaystyle 3$ and could be rewritten as $\displaystyle (x+2)$.
$\displaystyle -(x^2-2x+1)$ just means 'subtract everything inside the bracket.'
So we $\displaystyle -x^2$
we $\displaystyle -$ the $\displaystyle -2x$
and we $\displaystyle -1$
But because the $\displaystyle 2x$ is already negative, when we do the subtraction, it becomes positive.
If you're still confused, try considering $\displaystyle -2(3x+5)$
This can be rewritten as $\displaystyle -2(3x)+(-2)5$
$\displaystyle =-6x-10$
Can you follow that? Your example is extremely similar, except that we just have $\displaystyle -1(...)$
Ok, I've been working on the following expression to see if I could get the right result.
$\displaystyle (x-2)^2+4x$
So I first do:
$\displaystyle (x-2)(x-2)$
= $\displaystyle x^2-2x-2x-4+4x$
= $\displaystyle x^2-4$
So I think the result is this? $\displaystyle x^2-4$
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.