Question

How do I factor the trinomial 2a^2 + 30a + 100?

Answer 1

since all terms are even, you could start by factoring out a 2

Answer 2

It might get messy if you don't use the quadratic formula - well, actually, that might make things worse. But to start, you should cut the whole thing down by dividing by the first term's coefficient like so:
\[2a ^{2} +30a + 100 = a ^{2}+15a+50\]
From there, you can take the general form of a quadratic and treat it like so:

\[c ^{2}+2bc+b ^{2}\]
(Totally butchered the proper variables for that, I know, but whatever.)
Since you know that the...uhh, nope. Nevermind, it's not a square binomial. You'll just have to use the quadratic formula. But yeah, you should still divide by 2 first.

Answer 3

cant divide @mendicant_bias since its not an equation

Answer 4

Oh. Pardon, my stupid is showing. Well, then you'll just have to go all
\[x = \frac{ -b \pm \sqrt{b ^{2} - 4ac} }{ 2a } \]On that jawn't.

Answer 5

So do you get how to do it from here?

Answer 6

(On another note, it is in fact that time of the day where I have to go shove nutrients in my facehole and what not. I'm out, will help if still confused later.)

Answer 7

still wrong, you cannot solve for x

its not an equation

Answer 8

2a^2 + 30a + 100
factoring out a 2

2(a^2 + 15a + 50)

now we want to factor the trinomial within the parenthesis

a^2 + 15a + 50

(a+n)(a+m)= a^2+(n+m)a+nm = a^2 + 15a + 50

15= n+m
nm=50

solve for m and n

then substitute back into
(a+n)(a+m)

final solution will look something like
2*(a+n)(a+m)

Answer 9

What? I thought you literally just said that you can't divide by two.

Answer 10

In factoring out the two, aren't you doing the same if you don't-oh, just saw the last part. apologies.

Answer 11

And i'm not sure what you mean when you say "you cannot solve for x; it's not an equation." What distinguishes this from any other trinomial when you're solving for roots? Why is the quadratic formula not applicable here? It looks like it's a quadratic.

Answer 12

@Mertsj @Directrix @jim_thompson5910 @phi Could anybody explain to me what he's talking about? I trust he's right, but I don't get it; whenever you're solving for roots in a quadratic equation, the other side equals zero. This looks like it's a quadratic. The other side equals zero. Why can you not apply the quadratic formula?

Answer 13

@precal @Luis_Rivera @mathstudent55 @mukushla Anybody?

Answer 14

if you want to factor ax^2 + bx + c, you can temporarily set it equal to zero to get

ax^2 + bx + c = 0

then solve for x to get the two roots

x = p
x = q

From there, you can factor like so

x = p or x = q

x-p = 0 or x-q = 0

(x-p)(x-q) = 0


So the factorization of ax^2 + bx + c is (x-p)(x-q) where p and q are the two roots of ax^2 + bx + c

Answer 15

oh wait, this trick only works if a = 1, well you can still scale things, so it still works out in a way

Answer 16

I understand that, but why is that not applicable in this situation? He said that you can't use the quadratic formula since it's "not an equation". Was he just incorrect from you know, or what? I don't get it.

Answer 17

*from what you know, or what?

Answer 18

well technically it's not an equation, so the quadratic formula wouldn't apply

but you can still use the quadratic formula to find the roots, which will lead you to the factorization

Answer 19

the big thing to worry about is if a isn't equal to 1, then you would have to scale the equation so things match up

Answer 20

Like just having a coefficient outside of the quadratic fomula, or how would you do that?

Answer 21

Oh well. I think i'm just going to put this to rest. Thanks.

Answer 22

yes I think that's the trick but I'm trying to see if there are any examples where it's not so easy

Answer 23

In this case, 2a^2 + 30a + 100 has roots of x = -5 or x = -10 when you use the quadratic formula

so

x = -5 or x = -10

x+5 = 0 or x+10 = 0

(x+5)(x+10) = 0

then you multiply both sides by 2 to get

2(x+5)(x+10) = 0

so that works because that is the correct factorization of 2a^2 + 30a + 100

Answer 24

Okay, sweet. That explains a lot more.