SOLUTION: Please help me answer this question. What are the linear factors of 12x^2+7x-12? The answer given is 4x-3 and 3x+4. I'm just not too sure how to get to this answer.

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Question 276480: Please help me answer this question.
What are the linear factors of 12x^2+7x-12?
The answer given is 4x-3 and 3x+4.
I'm just not too sure how to get to this answer.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


Looking at the expression 12x%5E2%2B7x-12, we can see that the first coefficient is 12, the second coefficient is 7, and the last term is -12.



Now multiply the first coefficient 12 by the last term -12 to get %2812%29%28-12%29=-144.



Now the question is: what two whole numbers multiply to -144 (the previous product) and add to the second coefficient 7?



To find these two numbers, we need to list all of the factors of -144 (the previous product).



Factors of -144:

1,2,3,4,6,8,9,12,16,18,24,36,48,72,144

-1,-2,-3,-4,-6,-8,-9,-12,-16,-18,-24,-36,-48,-72,-144



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



These factors pair up and multiply to -144.

1*(-144) = -144
2*(-72) = -144
3*(-48) = -144
4*(-36) = -144
6*(-24) = -144
8*(-18) = -144
9*(-16) = -144
12*(-12) = -144
(-1)*(144) = -144
(-2)*(72) = -144
(-3)*(48) = -144
(-4)*(36) = -144
(-6)*(24) = -144
(-8)*(18) = -144
(-9)*(16) = -144
(-12)*(12) = -144


Now let's add up each pair of factors to see if one pair adds to the middle coefficient 7:



First NumberSecond NumberSum
1-1441+(-144)=-143
2-722+(-72)=-70
3-483+(-48)=-45
4-364+(-36)=-32
6-246+(-24)=-18
8-188+(-18)=-10
9-169+(-16)=-7
12-1212+(-12)=0
-1144-1+144=143
-272-2+72=70
-348-3+48=45
-436-4+36=32
-624-6+24=18
-818-8+18=10
-916-9+16=7
-1212-12+12=0




From the table, we can see that the two numbers -9 and 16 add to 7 (the middle coefficient).



So the two numbers -9 and 16 both multiply to -144 and add to 7



Now replace the middle term 7x with -9x%2B16x. Remember, -9 and 16 add to 7. So this shows us that -9x%2B16x=7x.



12x%5E2%2Bhighlight%28-9x%2B16x%29-12 Replace the second term 7x with -9x%2B16x.



%2812x%5E2-9x%29%2B%2816x-12%29 Group the terms into two pairs.



3x%284x-3%29%2B%2816x-12%29 Factor out the GCF 3x from the first group.



3x%284x-3%29%2B4%284x-3%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-3%29 Combine like terms. Or factor out the common term 4x-3



===============================================================



Answer:



So 12%2Ax%5E2%2B7%2Ax-12 factors to %283x%2B4%29%284x-3%29.



In other words, 12%2Ax%5E2%2B7%2Ax-12=%283x%2B4%29%284x-3%29.



Note: you can check the answer by expanding %283x%2B4%29%284x-3%29 to get 12%2Ax%5E2%2B7%2Ax-12 or by graphing the original expression and the answer (the two graphs should be identical).