SOLUTION: factoring a quadratic with leading coefficient greater than one thank you I need 3 question answer, please help I need them before time runs out 6y^2+y-12 thank you so ver

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Question 91474: factoring a quadratic with leading coefficient greater than one
thank you I need 3 question answer, please help I need them before time runs out
6y^2+y-12
thank you so very much
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 6y%5E2%2By-12, we can see that the first coefficient is 6, the second coefficient is 1, and the last term is -12.



Now multiply the first coefficient 6 by the last term -12 to get %286%29%28-12%29=-72.



Now the question is: what two whole numbers multiply to -72 (the previous product) and add to the second coefficient 1?



To find these two numbers, we need to list all of the factors of -72 (the previous product).



Factors of -72:

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

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



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



These factors pair up and multiply to -72.

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


Now let's add up each pair of factors to see if one pair adds to the middle coefficient 1:



First NumberSecond NumberSum
1-721+(-72)=-71
2-362+(-36)=-34
3-243+(-24)=-21
4-184+(-18)=-14
6-126+(-12)=-6
8-98+(-9)=-1
-172-1+72=71
-236-2+36=34
-324-3+24=21
-418-4+18=14
-612-6+12=6
-89-8+9=1




From the table, we can see that the two numbers -8 and 9 add to 1 (the middle coefficient).



So the two numbers -8 and 9 both multiply to -72 and add to 1



Now replace the middle term 1y with -8y%2B9y. Remember, -8 and 9 add to 1. So this shows us that -8y%2B9y=1y.



6y%5E2%2Bhighlight%28-8y%2B9y%29-12 Replace the second term 1y with -8y%2B9y.



%286y%5E2-8y%29%2B%289y-12%29 Group the terms into two pairs.



2y%283y-4%29%2B%289y-12%29 Factor out the GCF 2y from the first group.



2y%283y-4%29%2B3%283y-4%29 Factor out 3 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.



%282y%2B3%29%283y-4%29 Combine like terms. Or factor out the common term 3y-4



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Answer:



So 6%2Ay%5E2%2By-12 factors to %282y%2B3%29%283y-4%29.



In other words, 6%2Ay%5E2%2By-12=%282y%2B3%29%283y-4%29.



Note: you can check the answer by expanding %282y%2B3%29%283y-4%29 to get 6%2Ay%5E2%2By-12 or by graphing the original expression and the answer (the two graphs should be identical).