SOLUTION: 8b^2+24b+18

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Question 174349: 8b^2+24b+18
Answer by jim_thompson5910(35256) About Me  (Show Source):
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8b%5E2%2B24b%2B18 Start with the given expression


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


Now let's focus on the inner expression 4b%5E2%2B12b%2B9




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Looking at 4b%5E2%2B12b%2B9 we can see that the first term is 4b%5E2 and the last term is 9 where the coefficients are 4 and 9 respectively.

Now multiply the first coefficient 4 and the last coefficient 9 to get 36. Now what two numbers multiply to 36 and add to the middle coefficient 12? Let's list all of the factors of 36:



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

-1,-2,-3,-4,-6,-9,-12,-18 ...List the negative factors as well. This will allow us to find all possible combinations

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

note: remember two negative numbers multiplied together make a positive number


Now which of these pairs add to 12? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 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 this list we can see that 6 and 6 add up to 12 and multiply to 36


Now looking at the expression 4b%5E2%2B12b%2B9, replace 12b with 6b%2B6b (notice 6b%2B6b adds up to 12b. So it is equivalent to 12b)

4b%5E2%2Bhighlight%286b%2B6b%29%2B9


Now let's factor 4b%5E2%2B6b%2B6b%2B9 by grouping:


%284b%5E2%2B6b%29%2B%286b%2B9%29 Group like terms


2b%282b%2B3%29%2B3%282b%2B3%29 Factor out the GCF of 2b out of the first group. Factor out the GCF of 3 out of the second group


%282b%2B3%29%282b%2B3%29 Since we have a common term of 2b%2B3, we can combine like terms

So 4b%5E2%2B6b%2B6b%2B9 factors to %282b%2B3%29%282b%2B3%29


So this also means that 4b%5E2%2B12b%2B9 factors to %282b%2B3%29%282b%2B3%29 (since 4b%5E2%2B12b%2B9 is equivalent to 4b%5E2%2B6b%2B6b%2B9)


note: %282b%2B3%29%282b%2B3%29 is equivalent to %282b%2B3%29%5E2 since the term 2b%2B3 occurs twice. So 4b%5E2%2B12b%2B9 also factors to %282b%2B3%29%5E2



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So our expression goes from 2%284b%5E2%2B12b%2B9%29 and factors further to 2%282b%2B3%29%5E2


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Answer:

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