SOLUTION: 8b^2+24b+18

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Question 392932: 8b^2+24b+18
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
I'm assuming you want to factor.


8b%5E2%2B24b%2B18 Start with the given expression.


2%284b%5E2%2B12b%2B9%29 Factor out the GCF 2.


Now let's try to factor the inner expression 4b%5E2%2B12b%2B9


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Looking at the expression 4b%5E2%2B12b%2B9, we can see that the first coefficient is 4, the second coefficient is 12, and the last term is 9.


Now multiply the first coefficient 4 by the last term 9 to get %284%29%289%29=36.


Now the question is: what two whole numbers multiply to 36 (the previous product) and add to the second coefficient 12?


To find these two numbers, we need to list all of the factors of 36 (the previous product).


Factors of 36:
1,2,3,4,6,9,12,18,36
-1,-2,-3,-4,-6,-9,-12,-18,-36


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


These factors pair up and multiply to 36.
1*36 = 36
2*18 = 36
3*12 = 36
4*9 = 36
6*6 = 36
(-1)*(-36) = 36
(-2)*(-18) = 36
(-3)*(-12) = 36
(-4)*(-9) = 36
(-6)*(-6) = 36

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 12:


First NumberSecond NumberSum
1361+36=37
2182+18=20
3123+12=15
494+9=13
666+6=12
-1-36-1+(-36)=-37
-2-18-2+(-18)=-20
-3-12-3+(-12)=-15
-4-9-4+(-9)=-13
-6-6-6+(-6)=-12



From the table, we can see that the two numbers 6 and 6 add to 12 (the middle coefficient).


So the two numbers 6 and 6 both multiply to 36 and add to 12


Now replace the middle term 12b with 6b%2B6b. Remember, 6 and 6 add to 12. So this shows us that 6b%2B6b=12b.


4b%5E2%2Bhighlight%286b%2B6b%29%2B9 Replace the second term 12b with 6b%2B6b.


%284b%5E2%2B6b%29%2B%286b%2B9%29 Group the terms into two pairs.


2b%282b%2B3%29%2B%286b%2B9%29 Factor out the GCF 2b from the first group.


2b%282b%2B3%29%2B3%282b%2B3%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.


%282b%2B3%29%282b%2B3%29 Combine like terms. Or factor out the common term 2b%2B3


%282b%2B3%29%5E2 Condense the terms.


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So 2%284b%5E2%2B12b%2B9%29 then factors further to 2%282b%2B3%29%5E2


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Answer:


So 8b%5E2%2B24b%2B18 completely factors to 2%282b%2B3%29%5E2.


In other words, 8b%5E2%2B24b%2B18=2%282b%2B3%29%5E2.


Note: you can check the answer by expanding 2%282b%2B3%29%5E2 to get 8b%5E2%2B24b%2B18 or by graphing the original expression and the answer (the two graphs should be identical).


If you need more help, email me at jim_thompson5910@hotmail.com

Also, feel free to check out my tutoring website

Jim