SOLUTION: Please solve this problem and show the steps in solving it. 12y^2 - 5y = 2.

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Question 282331: Please solve this problem and show the steps in solving it. 12y^2 - 5y = 2.
Found 2 solutions by mananth, richwmiller:
Answer by mananth(16946) About Me  (Show Source):
You can put this solution on YOUR website!
12y^2-5y=2
12y^2-5y-2=0
12y^2-8y+3y-2
4y(3y-2)+1(3y-2)=0
(3y-2)(4y+1)=0
If 3y-2 = 0
3y=2
y=2/3
OR
If 4y+1=0
4y=-1
y=-1/4

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)


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



Now multiply the first coefficient 12 by the last term -2 to get %2812%29%28-2%29=-24.



Now the question is: what two whole numbers multiply to -24 (the previous product) and add to the second coefficient -5?



To find these two numbers, we need to list all of the factors of -24 (the previous product).



Factors of -24:

1,2,3,4,6,8,12,24

-1,-2,-3,-4,-6,-8,-12,-24



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



These factors pair up and multiply to -24.

1*(-24) = -24
2*(-12) = -24
3*(-8) = -24
4*(-6) = -24
(-1)*(24) = -24
(-2)*(12) = -24
(-3)*(8) = -24
(-4)*(6) = -24


Now let's add up each pair of factors to see if one pair adds to the middle coefficient -5:



First NumberSecond NumberSum
1-241+(-24)=-23
2-122+(-12)=-10
3-83+(-8)=-5
4-64+(-6)=-2
-124-1+24=23
-212-2+12=10
-38-3+8=5
-46-4+6=2




From the table, we can see that the two numbers 3 and -8 add to -5 (the middle coefficient).



So the two numbers 3 and -8 both multiply to -24 and add to -5



Now replace the middle term -5y with 3y-8y. Remember, 3 and -8 add to -5. So this shows us that 3y-8y=-5y.



12y%5E2%2Bhighlight%283y-8y%29-2 Replace the second term -5y with 3y-8y.



%2812y%5E2%2B3y%29%2B%28-8y-2%29 Group the terms into two pairs.



3y%284y%2B1%29%2B%28-8y-2%29 Factor out the GCF 3y from the first group.



3y%284y%2B1%29-2%284y%2B1%29 Factor out 2 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.



%283y-2%29%284y%2B1%29 Combine like terms. Or factor out the common term 4y%2B1



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Answer:



So 12%2Ay%5E2-5%2Ay-2 factors to %283y-2%29%284y%2B1%29.



In other words, 12%2Ay%5E2-5%2Ay-2=%283y-2%29%284y%2B1%29.



Note: you can check the answer by expanding %283y-2%29%284y%2B1%29 to get 12%2Ay%5E2-5%2Ay-2 or by graphing the original expression and the answer (the two graphs should be identical).