SOLUTION: Factor. 2y^2+17y+30

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Question 276797: Factor.
2y^2+17y+30
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Looking at the expression 2y%5E2%2B17y%2B30, we can see that the first coefficient is 2, the second coefficient is 17, and the last term is 30.


Now multiply the first coefficient 2 by the last term 30 to get %282%29%2830%29=60.


Now the question is: what two whole numbers multiply to 60 (the previous product) and add to the second coefficient 17?


To find these two numbers, we need to list all of the factors of 60 (the previous product).


Factors of 60:
1,2,3,4,5,6,10,12,15,20,30,60
-1,-2,-3,-4,-5,-6,-10,-12,-15,-20,-30,-60


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


These factors pair up and multiply to 60.
1*60 = 60
2*30 = 60
3*20 = 60
4*15 = 60
5*12 = 60
6*10 = 60
(-1)*(-60) = 60
(-2)*(-30) = 60
(-3)*(-20) = 60
(-4)*(-15) = 60
(-5)*(-12) = 60
(-6)*(-10) = 60

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 17:


First NumberSecond NumberSum
1601+60=61
2302+30=32
3203+20=23
4154+15=19
5125+12=17
6106+10=16
-1-60-1+(-60)=-61
-2-30-2+(-30)=-32
-3-20-3+(-20)=-23
-4-15-4+(-15)=-19
-5-12-5+(-12)=-17
-6-10-6+(-10)=-16



From the table, we can see that the two numbers 5 and 12 add to 17 (the middle coefficient).


So the two numbers 5 and 12 both multiply to 60 and add to 17


Now replace the middle term 17y with 5y%2B12y. Remember, 5 and 12 add to 17. So this shows us that 5y%2B12y=17y.


2y%5E2%2Bhighlight%285y%2B12y%29%2B30 Replace the second term 17y with 5y%2B12y.


%282y%5E2%2B5y%29%2B%2812y%2B30%29 Group the terms into two pairs.


y%282y%2B5%29%2B%2812y%2B30%29 Factor out the GCF y from the first group.


y%282y%2B5%29%2B6%282y%2B5%29 Factor out 6 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%2B6%29%282y%2B5%29 Combine like terms. Or factor out the common term 2y%2B5


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Answer:


So 2y%5E2%2B17y%2B30 factors to %28y%2B6%29%282y%2B5%29.


In other words, 2y%5E2%2B17y%2B30=%28y%2B6%29%282y%2B5%29.


Note: you can check the answer by expanding %28y%2B6%29%282y%2B5%29 to get 2y%5E2%2B17y%2B30 or by graphing the original expression and the answer (the two graphs should be identical).