SOLUTION: 36x^2+24x+4

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Question 185047: 36x^2+24x+4
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
I assume that you want to factor.


36x%5E2%2B24x%2B4 Start with the given expression


4%289x%5E2%2B6x%2B1%29 Factor out the GCF 4


Now let's focus on the inner expression 9x%5E2%2B6x%2B1




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Looking at 9x%5E2%2B6x%2B1 we can see that the first term is 9x%5E2 and the last term is 1 where the coefficients are 9 and 1 respectively.

Now multiply the first coefficient 9 and the last coefficient 1 to get 9. Now what two numbers multiply to 9 and add to the middle coefficient 6? Let's list all of the factors of 9:



Factors of 9:
1,3

-1,-3 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to 9
1*9
3*3
(-1)*(-9)
(-3)*(-3)

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


Now which of these pairs add to 6? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 6

First NumberSecond NumberSum
191+9=10
333+3=6
-1-9-1+(-9)=-10
-3-3-3+(-3)=-6



From this list we can see that 3 and 3 add up to 6 and multiply to 9


Now looking at the expression 9x%5E2%2B6x%2B1, replace 6x with 3x%2B3x (notice 3x%2B3x adds up to 6x. So it is equivalent to 6x)

9x%5E2%2Bhighlight%283x%2B3x%29%2B1


Now let's factor 9x%5E2%2B3x%2B3x%2B1 by grouping:


%289x%5E2%2B3x%29%2B%283x%2B1%29 Group like terms


3x%283x%2B1%29%2B1%283x%2B1%29 Factor out the GCF of 3x out of the first group. Factor out the GCF of 1 out of the second group


%283x%2B1%29%283x%2B1%29 Since we have a common term of 3x%2B1, we can combine like terms

So 9x%5E2%2B3x%2B3x%2B1 factors to %283x%2B1%29%283x%2B1%29


So this also means that 9x%5E2%2B6x%2B1 factors to %283x%2B1%29%283x%2B1%29 (since 9x%5E2%2B6x%2B1 is equivalent to 9x%5E2%2B3x%2B3x%2B1)


note: %283x%2B1%29%283x%2B1%29 is equivalent to %283x%2B1%29%5E2 since the term 3x%2B1 occurs twice. So 9x%5E2%2B6x%2B1 also factors to %283x%2B1%29%5E2



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So our expression goes from 4%289x%5E2%2B6x%2B1%29 and factors further to 4%283x%2B1%29%5E2


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Answer:

So 36x%5E2%2B24x%2B4 factors to 4%283x%2B1%29%5E2