SOLUTION: Factor completely. 6y^2 + 27y - 15

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Question 283377: Factor completely.
6y^2 + 27y - 15
Answer by Deina(147) About Me  (Show Source):
You can put this solution on YOUR website!
I've always hated factoring!
Thanks to Jim Thompson for writing this, it explains the process far better than I ever could!
Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


6%2Ay%5E2%2B27%2Ay-15 Start with the given expression.



3%282y%5E2%2B9y-5%29 Factor out the GCF 3.



Now let's try to factor the inner expression 2y%5E2%2B9y-5



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Looking at the expression 2y%5E2%2B9y-5, we can see that the first coefficient is 2, the second coefficient is 9, and the last term is -5.



Now multiply the first coefficient 2 by the last term -5 to get %282%29%28-5%29=-10.



Now the question is: what two whole numbers multiply to -10 (the previous product) and add to the second coefficient 9?



To find these two numbers, we need to list all of the factors of -10 (the previous product).



Factors of -10:

1,2,5,10

-1,-2,-5,-10



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



These factors pair up and multiply to -10.

1*(-10) = -10
2*(-5) = -10
(-1)*(10) = -10
(-2)*(5) = -10


Now let's add up each pair of factors to see if one pair adds to the middle coefficient 9:



First NumberSecond NumberSum
1-101+(-10)=-9
2-52+(-5)=-3
-110-1+10=9
-25-2+5=3




From the table, we can see that the two numbers -1 and 10 add to 9 (the middle coefficient).



So the two numbers -1 and 10 both multiply to -10 and add to 9



Now replace the middle term 9y with -y%2B10y. Remember, -1 and 10 add to 9. So this shows us that -y%2B10y=9y.



2y%5E2%2Bhighlight%28-y%2B10y%29-5 Replace the second term 9y with -y%2B10y.



%282y%5E2-y%29%2B%2810y-5%29 Group the terms into two pairs.



y%282y-1%29%2B%2810y-5%29 Factor out the GCF y from the first group.



y%282y-1%29%2B5%282y-1%29 Factor out 5 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%2B5%29%282y-1%29 Combine like terms. Or factor out the common term 2y-1



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So 3%282y%5E2%2B9y-5%29 then factors further to 3%28y%2B5%29%282y-1%29



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Answer:



So 6%2Ay%5E2%2B27%2Ay-15 completely factors to 3%28y%2B5%29%282y-1%29.



In other words, 6%2Ay%5E2%2B27%2Ay-15=3%28y%2B5%29%282y-1%29.



Note: you can check the answer by expanding 3%28y%2B5%29%282y-1%29 to get 6%2Ay%5E2%2B27%2Ay-15 or by graphing the original expression and the answer (the two graphs should be identical).


and bobs your uncle!