Question 892214: How do I factor
3x^2+5x+2
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! your equation is in the form of ax^2 + bx + c = 0
a = 3
b = 5
c = 2
your factors will be (a1x + c1) * (a2x + c2)
a1 and a2 are factors of 3, so they will be 3 * 1 or 1 * 3
c1 and c2 are factors of 2, so they will be 2 * 1 or 1 * 2
the middle term needs to be a combination of a1*c2 + a2*c1
since the b term is 5, then a1*c2 * a2*c1 needs to be equal to 5.
since 5 = 3 + 2, than a1c2 needs to be equal to 2 and a2c1 needs to be equal to 3
your possible combinations are 3*2 + 1*1 or 3*1 + 1*2
of these possible combinations, 3*1 + 1*2 looks the most promising.
so let a1 = 3 and c2 = 1 and let a2 = 1 and c1 = 2 and your factors of (a1x + c1) * (a2x + c2) become (3*x + 2) * (x + 1)
if you multiply these factors together, you will get 3x^2 + 3x + 2x + 2 which will be equal to 3x^2 + 5x + 2 after you combine like terms.
your factors are (3x+2)*(x+1)
there are several different methods you can choose.
another method is to split the middle term as follows:
your equation is 3x^2 + 5x + 2
multiply the coefficiennt of the x^2 term by the constant term to get 3*2 = 6
now look for all possible factors of 6 until you find a set of factors that will add up to 5.
1*6 = 6 but doesn't add up to 5
2*3 = 6 and does add up to 5.
thoe are your factors.
now split the middle term into 2x and 3x because they add up to 5xc.
you will get:
3x^2 + 2x + 3x + 2
now associate the first 2 terms together and the last 2 terms together to get:
(3x^2 + 2x) + (3x + 2)
now factor the left term and the right term as much as you can.
if your left term is left with a factor of 3x+2, you want to get your right term to have a factor of 3x+2 as well.
it should happen naturally, as it does in this case.
you will get x*(3x+2) + 1*(3x+2)
since you have a common factor of (3x+2), you can factor it out to get:
(x+1)*(3x+2)
there is another method called the box method which is similar to this last method but uses a box and is a little more formal in the step as hey are carried out.
all of these methods can be seen online.
some good references for you to look at are shown below:
the method of last resort is the quadratic formula.
if you can't factor the equation any other way, you can always factor it using the quadratic formula.
that's one of the references below as well.
there are more, including a new method i just learned about that i already forgot and the indian method which is an offshoot of the methods referenced below.
for practical purposes, if you learn the box method and the quadratic formula, you should be in good shape.
the splitting the b term method i showed you is an offshoot of the box method but a little less formal and a little harder to understand because of the factoring out of the common terms isn't as formalized as in the box method.
http://www.purplemath.com/modules/factquad2.htm
http://www.purplemath.com/modules/factquad.htm
http://www.purplemath.com/modules/factquad.htm
http://www.purplemath.com/modules/quadform.htm
|
|
|
