Question

How do I solve 10p^2-5pq-180q^2??

Answer 1

You can't solve something if you don't have an = in there somewhere.

Answer 2

I cant figure out how to factor it.

Answer 3

Well, do you see any common factors?

Answer 4

yes,5.

Answer 5

Great! So what does it look like after you factor out the 5?

Answer 6

2p^2-pq-36q^2

Answer 7

Well, \[5(2p^2-pq-36q^2)\]right?

Let's set the 5 aside for the moment and factor the trinomial in the parentheses. First, a demonstration:

\[(x+a)(x+b) = x^2 + bx + ax + ab = x^2 + (a+b)x + ab\]
Notice how the final term comes entirely from the product of the right hand portion of the two binomials, and the first term comes entirely from the product of the left hand portion of the two binomials?

Answer 8

Given that, how do you think we are going to get a \(2p^2\) term by multiplying two binomials? It won't be \((p+a)(p+b)\) because that would only give us \(p^2\). Any ideas?

Answer 9

OK,I'm lost.I have no idea.

Answer 10

What two things could we multiply together to get \(2p^2\)?

Answer 11

Let's say that one of them is \(p\). What would the other be?

Answer 12

p and 2p?

Answer 13

Right! As it turns out, those are our only options. If we had \(4p^2\), it could be \( p*4p \) or it could be \(2p*2p\). So that means our factored trinomial is going to be something along the lines of
\[(2p + a)(p + b)\] and we still have to work out the values of a and b (which may be negative).

Answer 14

Now let's look at the final term of the trinomial: \(-36q^2\)

Any ideas on how we'll construct that? There's going to be a \(q*q\) portion to produce the \(q^2\), and then two numbers which will give -36 as a product.

Answer 15

That means we'll have something like\[(2p+aq)(p+bq)\] and a*b = -36 because if we expand what we have so far,
\[(2p+aq)(p+bq) = 2p^2+2bpq + apq + abq^2 = 2p^2 + (2b+a)pq + abq^2\]
Matching up like terms with the trinomial, we see that (2b+a) = -1 and ab = -36.

Answer 16

2q and 18q

Answer 17

If we didn't have that pesky 2 in front of the p^2 term, that might be right.

So we need the two numbers that when multiplied = -36, and when one them is doubled and added to the other, you get -1. Clearly, one is positive and one is negative because the product is negative, and the only way you get a negative product is to have an odd number of negative numbers being multiplied.

Answer 18

Let's have a look at the factors of -36:
-1, 36
-2, 18
-3, 12
-4, 9
-6, 6
-9, 4
-12, 3
-18, 2
-36, 1

Do any of those pairs satisfy our needs? Doubling one number and adding the other has to give us -1 as a result.

Answer 19

I dont see a pair that has a sum of -1.

Answer 20

No, it's not a pair that has a sum of -1, it's a pair that when you double one and add the other, it's a sum of -1. Here's a hint: the number that gets doubled is not in the first column...

Answer 21

I dont understand that.

Answer 22

Are you not seeing which one works, or do you not understand why I'm saying that we double one and add the other?

Answer 23

I dont understand why we doubel

Answer 24

Okay. You're in agreement that our factored trinomial will look like
(2p stuff)(p stuff)?

Answer 25

We need 2p * p to make 2p^2 in the first term.

Answer 26

OK.I understand that.

Answer 27

Okay, well, that means that the "stuff" in the second part gets multiplied by 2p, not p.

Say we had a simpler polynomial only in terms of \(p\), and we had already factored it to get
\[(2p-4)(p+3)\]If we expand that (multiply it out), we get
\[2p*p + 3*2p -4*p -4*3 = 2p^2+6p-4p-12 = 2p^2+2p-12\]
Notice that the +3 got multiplied by 2p, not 1p, so it contributed a 6p instead of a 3p to the sum that makes up the p term?

Answer 28

If we had to go the other way, we'd be looking at
\[2p^2 + 2p - 12 = (2p + a)(p + b)\] and trying to figure out which pair of factors of -12 when you double one and add the other gives you 2. Does that make sense?

Answer 29

Ok i sorta understand it now.

Answer 30

I agree, it takes some getting used to...

Going to our original problem, we wanted two factors of -36 that when we double one and add the other, we get -1. What about 4 and -9? -9*4 = -36, and 2*4-9=-1. What is our factored trinomial? It will be either
\[(2p+4)(p-9)\]or\[(2p-9)(p+4)\]

Answer 31

(I'm leaving out the \(q\) portion for the moment, just to make it a bit clearer)

Answer 32

Multiply them both out and see which one is correct...

Answer 33

@DogLuverSarah2012
What's your choice?