document.write( "Question 91474: factoring a quadratic with leading coefficient greater than one\r
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Algebra.Com's Answer #66423 by jim_thompson5910(35256)\"\" \"About 
You can put this solution on YOUR website!
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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\":



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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).

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