SOLUTION: Factor. 12x^2+xy-6y^2

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Question 333713: Factor.
12x^2+xy-6y^2
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Looking at the expression 12x%5E2%2Bxy-6y%5E2, we can see that the first coefficient is 12, the second coefficient is 1, and the last coefficient is -6.


Now multiply the first coefficient 12 by the last coefficient -6 to get %2812%29%28-6%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 1xy with -8xy%2B9xy. Remember, -8 and 9 add to 1. So this shows us that -8xy%2B9xy=1xy.


12x%5E2%2Bhighlight%28-8xy%2B9xy%29-6y%5E2 Replace the second term 1xy with -8xy%2B9xy.


%2812x%5E2-8xy%29%2B%289xy-6y%5E2%29 Group the terms into two pairs.


4x%283x-2y%29%2B%289xy-6y%5E2%29 Factor out the GCF 4x from the first group.


4x%283x-2y%29%2B3y%283x-2y%29 Factor out 3y 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.


%284x%2B3y%29%283x-2y%29 Combine like terms. Or factor out the common term 3x-2y


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Answer:


So 12x%5E2%2Bxy-6y%5E2 factors to %284x%2B3y%29%283x-2y%29.


In other words, 12x%5E2%2Bxy-6y%5E2=%284x%2B3y%29%283x-2y%29.


Note: you can check the answer by expanding %284x%2B3y%29%283x-2y%29 to get 12x%5E2%2Bxy-6y%5E2 or by graphing the original expression and the answer (the two graphs should be identical).