1. ## expanding binomials

Promise no more questions today! ^_^

The reason I'm stuck on these is because I've completely forgotten what im doing, after returning to these from months ago.

I need to:

1) Express $\displaystyle x^2 - 8x + 15$ in the form $\displaystyle (x + a )^2 + b$ where "a" and "b" are the numbers to be found.

This is what I've got thus far:

$\displaystyle y = x^2 - 8x + 15$

$\displaystyle = (x - 4 )^2 - (4)^2 + 15 = 0$

$\displaystyle (x - 4)^2 - 16 + 15$

Then I need to plot for the original equation. Not alot

2) Expand the following using Pascal's Triangle

$\displaystyle (2 + x)^3$

I could manage the $\displaystyle (x+1)^n$ expansion but not sure how to go about this one.

Thanks

2. Hello,

Originally Posted by Nick87
Promise no more questions today! ^_^

The reason I'm stuck on these is because I've completely forgotten what im doing, after returning to these from months ago.

I need to:

1) Express $\displaystyle x^2 - 8x + 15$ in the form $\displaystyle (x + a )^2 + b$ where "a" and "b" are the numbers to be found.

This is what I've got thus far:

$\displaystyle y = x^2 - 8x + 15$

$\displaystyle = (x - 4 )^2 - (4)^2 + 15 = 0$

$\displaystyle (x - 4)^2 - 16 + 15$

Then I need to plot for the original equation. Not alot
Sorry if I misunderstood what you've said, but you got it

2) Expand the following using Pascal's Triangle

$\displaystyle (2 + x)^3$

I could manage the $\displaystyle (x+1)^n$ expansion but not sure how to go about this one.

Thanks
So you know the formula for $\displaystyle (x+1)^n$

It's the same for $\displaystyle (2+x)^3$

$\displaystyle =C(3,0) 2^0 x^3+C(3,1)2^1 x^2+C(3,2) 2^2 x^1+C(3,3) 2^3 x^0$

Where $\displaystyle C(n,p)={p \choose n}$

3. So for the first one $\displaystyle a=4$ and $\displaystyle b=-16$?

It says to use my answer to sketch the graph of $\displaystyle y = x^2 - 8x + 15$ so what does that mean? Can't I just draw the graph like usual?

4. Originally Posted by Nick87
So for the first one $\displaystyle a=4$ and $\displaystyle b=-16$?

It says to use my answer to sketch the graph of $\displaystyle y = x^2 - 8x + 15$ so what does that mean? Can't I just draw the graph like usual?
Erm...

$\displaystyle y=(x-4)^2-16+15=(x-4)^2-1$

--> $\displaystyle a=-4$ and $\displaystyle b=-1$

To graph the thing, represent yourself :

- you know how to graph y=x²
- imagine that if you add a certain number to x, it's like translating to the right or the left the graph
- adding a constant (here, b) is like increasing or decreasing the curve, that is to say "go up or go down", because you add b to the ordinate of each point

5. Originally Posted by Moo
Erm...

$\displaystyle y=(x-4)^2-16+15=(x-4)^2-1$

--> $\displaystyle a=-4$ and $\displaystyle b=-1$
*Slaps self* of course its -1!

So what I need to do now is plot the orignal graph but with the translation $\displaystyle {a \choose b}$?

6. Originally Posted by Nick87
*Slaps self* of course its -1!

So what I need to do now is plot the orignal graph but with the translation $\displaystyle {a \choose b}$?
Yep !
I have an hesitation, perhaps it's -a. Check on the graph which one it is

7. Thank you so much for all your help!