SOLUTION: 8y^2+10y+3=

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Question 251571: 8y^2+10y+3=
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Looking at the expression 8y%5E2%2B10y%2B3, we can see that the first coefficient is 8, the second coefficient is 10, and the last term is 3.


Now multiply the first coefficient 8 by the last term 3 to get %288%29%283%29=24.


Now the question is: what two whole numbers multiply to 24 (the previous product) and add to the second coefficient 10?


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 10:


First NumberSecond NumberSum
1241+24=25
2122+12=14
383+8=11
464+6=10
-1-24-1+(-24)=-25
-2-12-2+(-12)=-14
-3-8-3+(-8)=-11
-4-6-4+(-6)=-10



From the table, we can see that the two numbers 4 and 6 add to 10 (the middle coefficient).


So the two numbers 4 and 6 both multiply to 24 and add to 10


Now replace the middle term 10y with 4y%2B6y. Remember, 4 and 6 add to 10. So this shows us that 4y%2B6y=10y.


8y%5E2%2Bhighlight%284y%2B6y%29%2B3 Replace the second term 10y with 4y%2B6y.


%288y%5E2%2B4y%29%2B%286y%2B3%29 Group the terms into two pairs.


4y%282y%2B1%29%2B%286y%2B3%29 Factor out the GCF 4y from the first group.


4y%282y%2B1%29%2B3%282y%2B1%29 Factor out 3 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.


%284y%2B3%29%282y%2B1%29 Combine like terms. Or factor out the common term 2y%2B1


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Answer:


So 8y%5E2%2B10y%2B3 factors to %284y%2B3%29%282y%2B1%29.


In other words, 8y%5E2%2B10y%2B3=%284y%2B3%29%282y%2B1%29.


Note: you can check the answer by expanding %284y%2B3%29%282y%2B1%29 to get 8y%5E2%2B10y%2B3 or by graphing the original expression and the answer (the two graphs should be identical).