SOLUTION: Factor: 12a^2 + 5a - 2.

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Question 113856: Factor: 12a^2 + 5a - 2.
Answer by jim_thompson5910(35256) About Me  (Show Source):
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Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


Looking at the expression 12a%5E2%2B5a-2, we can see that the first coefficient is 12, the second coefficient is 5, and the last term is -2.



Now multiply the first coefficient 12 by the last term -2 to get %2812%29%28-2%29=-24.



Now the question is: what two whole numbers multiply to -24 (the previous product) and add to the second coefficient 5?



To find these two numbers, we need to list all of the factors of -24 (the previous product).



Factors of -24:

1,2,3,4,6,8,12,24

-1,-2,-3,-4,-6,-8,-12,-24



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



These factors pair up and multiply to -24.

1*(-24) = -24
2*(-12) = -24
3*(-8) = -24
4*(-6) = -24
(-1)*(24) = -24
(-2)*(12) = -24
(-3)*(8) = -24
(-4)*(6) = -24


Now let's add up each pair of factors to see if one pair adds to the middle coefficient 5:



First NumberSecond NumberSum
1-241+(-24)=-23
2-122+(-12)=-10
3-83+(-8)=-5
4-64+(-6)=-2
-124-1+24=23
-212-2+12=10
-38-3+8=5
-46-4+6=2




From the table, we can see that the two numbers -3 and 8 add to 5 (the middle coefficient).



So the two numbers -3 and 8 both multiply to -24 and add to 5



Now replace the middle term 5a with -3a%2B8a. Remember, -3 and 8 add to 5. So this shows us that -3a%2B8a=5a.



12a%5E2%2Bhighlight%28-3a%2B8a%29-2 Replace the second term 5a with -3a%2B8a.



%2812a%5E2-3a%29%2B%288a-2%29 Group the terms into two pairs.



3a%284a-1%29%2B%288a-2%29 Factor out the GCF 3a from the first group.



3a%284a-1%29%2B2%284a-1%29 Factor out 2 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.



%283a%2B2%29%284a-1%29 Combine like terms. Or factor out the common term 4a-1



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Answer:



So 12%2Aa%5E2%2B5%2Aa-2 factors to %283a%2B2%29%284a-1%29.



In other words, 12%2Aa%5E2%2B5%2Aa-2=%283a%2B2%29%284a-1%29.



Note: you can check the answer by expanding %283a%2B2%29%284a-1%29 to get 12%2Aa%5E2%2B5%2Aa-2 or by graphing the original expression and the answer (the two graphs should be identical).