SOLUTION: Factor Completely: 4y^2-28y-72=

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Question 898407: Factor Completely:
4y^2-28y-72=
Answer by richwmiller(17219) About Me  (Show Source):
You can put this solution on YOUR website!
Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


4%2Ay%5E2-28%2Ay-72 Start with the given expression.



4%28y%5E2-7y-18%29 Factor out the GCF 4.



Now let's try to factor the inner expression y%5E2-7y-18



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Looking at the expression y%5E2-7y-18, we can see that the first coefficient is 1, the second coefficient is -7, and the last term is -18.



Now multiply the first coefficient 1 by the last term -18 to get %281%29%28-18%29=-18.



Now the question is: what two whole numbers multiply to -18 (the previous product) and add to the second coefficient -7?



To find these two numbers, we need to list all of the factors of -18 (the previous product).



Factors of -18:

1,2,3,6,9,18

-1,-2,-3,-6,-9,-18



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



These factors pair up and multiply to -18.

1*(-18) = -18
2*(-9) = -18
3*(-6) = -18
(-1)*(18) = -18
(-2)*(9) = -18
(-3)*(6) = -18


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



First NumberSecond NumberSum
1-181+(-18)=-17
2-92+(-9)=-7
3-63+(-6)=-3
-118-1+18=17
-29-2+9=7
-36-3+6=3




From the table, we can see that the two numbers 2 and -9 add to -7 (the middle coefficient).



So the two numbers 2 and -9 both multiply to -18 and add to -7



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



y%5E2%2Bhighlight%282y-9y%29-18 Replace the second term -7y with 2y-9y.



%28y%5E2%2B2y%29%2B%28-9y-18%29 Group the terms into two pairs.



y%28y%2B2%29%2B%28-9y-18%29 Factor out the GCF y from the first group.



y%28y%2B2%29-9%28y%2B2%29 Factor out 9 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.



%28y-9%29%28y%2B2%29 Combine like terms. Or factor out the common term y%2B2



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So 4%28y%5E2-7y-18%29 then factors further to 4%28y-9%29%28y%2B2%29



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Answer:



So 4%2Ay%5E2-28%2Ay-72 completely factors to 4%28y-9%29%28y%2B2%29.



In other words, 4%2Ay%5E2-28%2Ay-72=4%28y-9%29%28y%2B2%29.



Note: you can check the answer by expanding 4%28y-9%29%28y%2B2%29 to get 4%2Ay%5E2-28%2Ay-72 or by graphing the original expression and the answer (the two graphs should be identical).