SOLUTION: I need help factoring 12x squared+19x+4

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Question 204682This question is from textbook
: I need help factoring 12x squared+19x+4 This question is from textbook

Found 2 solutions by nerdybill, jim_thompson5910:
Answer by nerdybill(7384) About Me  (Show Source):
You can put this solution on YOUR website!
I need help factoring 12x squared+19x+4
.
Apply the "ac method":
We get
(3x + 4)(4x + 1)

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Looking at the expression 12x%5E2%2B19x%2B4, we can see that the first coefficient is 12, the second coefficient is 19, and the last term is 4.


Now multiply the first coefficient 12 by the last term 4 to get %2812%29%284%29=48.


Now the question is: what two whole numbers multiply to 48 (the previous product) and add to the second coefficient 19?


To find these two numbers, we need to list all of the factors of 48 (the previous product).


Factors of 48:
1,2,3,4,6,8,12,16,24,48
-1,-2,-3,-4,-6,-8,-12,-16,-24,-48


Note: list the negative of each factor. This will allow us to find all possible combinations.


These factors pair up and multiply to 48.
1*48
2*24
3*16
4*12
6*8
(-1)*(-48)
(-2)*(-24)
(-3)*(-16)
(-4)*(-12)
(-6)*(-8)

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 19:


First NumberSecond NumberSum
1481+48=49
2242+24=26
3163+16=19
4124+12=16
686+8=14
-1-48-1+(-48)=-49
-2-24-2+(-24)=-26
-3-16-3+(-16)=-19
-4-12-4+(-12)=-16
-6-8-6+(-8)=-14



From the table, we can see that the two numbers 3 and 16 add to 19 (the middle coefficient).


So the two numbers 3 and 16 both multiply to 48 and add to 19


Now replace the middle term 19x with 3x%2B16x. Remember, 3 and 16 add to 19. So this shows us that 3x%2B16x=19x.


12x%5E2%2Bhighlight%283x%2B16x%29%2B4 Replace the second term 19x with 3x%2B16x.


%2812x%5E2%2B3x%29%2B%2816x%2B4%29 Group the terms into two pairs.


3x%284x%2B1%29%2B%2816x%2B4%29 Factor out the GCF 3x from the first group.


3x%284x%2B1%29%2B4%284x%2B1%29 Factor out 4 from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.


%283x%2B4%29%284x%2B1%29 Combine like terms. Or factor out the common term 4x%2B1

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Answer:


So 12x%5E2%2B19x%2B4 factors to %283x%2B4%29%284x%2B1%29.


Note: you can check the answer by FOILing %283x%2B4%29%284x%2B1%29 to get 12x%5E2%2B19x%2B4 or by graphing the original expression and the answer (the two graphs should be identical).