1. ## factoring

Can someone help me understand how to factor these questions? I mean work out one of them step by step so, I can try to figure one out. Thanks a bunch!
1. Factor completely.
2x2 + 40x + 200
a) 2(x + 10)(x + 20)
b) 2(x + 20)2
c) (2x + 10)2
d) 2(x + 10)2
e) none of the above

2. Factor 4x2 - 20x + 25. Select the answer.
a) 2x-5
b) (2x-5)2
c) (2x+5)2
d) 2x+5
e) none of these

2. Originally Posted by starrynight
1. Factor completely.
2x2 + 40x + 200
a) 2(x + 10)(x + 20)
b) 2(x + 20)2
c) (2x + 10)2
d) 2(x + 10)2
e) none of the above
Well even if you can't manage to factor them you can multiply each of the answers out to see which one is correct.

$\displaystyle 2x^2 + 40x + 200$

First note that there is a common factor of 2 in each term:
$\displaystyle 2x^2 + 40x + 200 = 2(x^2 + 20x + 100)$
(so you should know automatically that c) can't be correct, right?)

Now let's take a look at $\displaystyle x^2 + 20x + 100$.

Take the leading coefficient (1) and multiply it by the constant term (100), in this case giving us 100.

Now list all possible pairs of factors of 100:
1, 100
2, 50
4, 25
5, 20
10, 10
20, 5
25, 4
50, 2
100, 1
(and we have all the negatives as well: -1, 100 etc.)

Now find out which of these pairs add up to be the linear coefficient (20). I get that 10 + 10 = 20.

So what you want to do is to write 20x = 10x + 10x in your quadratic. So we get
$\displaystyle 2x^2 + 40x + 200 = 2(x^2 + 20x + 100)$

$\displaystyle = 2(x^2 + 10x + 10x + 100)$

$\displaystyle = 2([x^2 + 10x] + [10x + 100])$

$\displaystyle = 2(x[x + 10] + 10[x + 10])$

$\displaystyle = 2(x + 10)(x + 10) = 2(x + 10)^2$

Try this approach on your second problem.

-Dan

I think the answer is none of the above. Am I right?

4. Originally Posted by starrynight
I think the answer is none of the above. Am I right?
As I said before you can find the correct answer simply by FOILing out each possible answer, so there should be no doubt in your mind.

No, you don't have it correct.
$\displaystyle 4x^2 - 20x + 25$

Using the same method as I posted earlier:
4 times 25 = 100

Factors of 100:
1, 100
2, 50
etc.

And as I said, don't forget about the double negative pairs like -2, -50 etc.

I get that -20 = -10 + -10, so put the quadratic in the form:
$\displaystyle 4x^2 - 20x + 25 = 4x^2 - 10x - 10x + 25$

$\displaystyle = (4x^2 - 10x) + (-10x + 25)$

and do your factoring from there.

-Dan