1. Hm... okay.

$(y-x)(y-x) = (y \times y) + (y \times -x) + (y \times -x) + (-x \times -x)$

From my above description,

Multiply y in the first bracket by the first y in the second bracket. This gives $y^2$
Then, multiply y in the first bracket by -x in the second bracket. This gives $-xy$
Then, multiply -x in the first bracket by y in the second bracket. This gives $-xy$
Finally, multiply -x in the first bracket by -x in the second bracket. This gives $x^2$

Then,

$(y \times y) + (y \times -x) + (y \times -x) + (-x \times -x) = y^2 - xy - xy + x^2$

Is this better?

2. Fantastic! :P

Thanks!

3. I have another question:

when do we know, when we CAN/MUST score out the 24 here:

$x^2+y^2-24 = 81 +24
x^2+ y^2 =105m^2$

Not only here but in every sum, I've learned it last year, but I didn't understand it.
There must be a rule for it or something?

4. Hm... ok, I'll give the details for this.

$x^2+y^2-24 = 81 +24$

Take note that this is not good.

It is originally:

$x^2+y^2-24 = 81$

Okay, what you need to know, is that everything that you do on one side, you will do the same thing on the other side.

If you add something, here 24, you add 24 on both sides:

$x^2+y^2-24+24 = 81+24$

What do you see? -24 + 24 = 0

So, this becomes:

$x^2+y^2+0= 105$

$x^2+y^2= 105$

Let's see why.
5 = 5

This is correct, right?
If you add something, you add it to both sides so that the equation remains true.
5 +2 = 5 + 2
7 = 7

The same thing applies to subtraction, multiplication and division.

Is this okay?

5. I understand this.
But WHEN can we do this? Can we do this ALWAYS, like this:

maybe, if all the cyphers have lettres except 1. like 24?

6. You can always do this when you have an equation.

for example, if you are only given to find x + 5, using the fact that x = 1.

You cannot say x + 5 = 0, then x = -5.

You need to first put the value of x, then you evaluate the answer. That is (1) + 5 = 6.

But, if you want to use the equation itself...

x = 1

You need x + 5... so add 5 on both sides:

x + 5 = 1 + 5

x + 5 = 6

I'm telling you this because, often, you have for example, simplify $5(x +3) + 3(2+x)$

You cannot say that $5(x +3) + 2(3+x) = 0$

then say that x = 3 because $5(x +3) + 3(2+x)$ is an expression and $5(x +3) + 2(3+x) = 0$ is an equation.

The rule is: In an equation, if you do something on one side, do the same on the other side.

Did that help?

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