SOLUTION: i can't figure out how to factor 4x^2-9+2

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Question 225931: i can't figure out how to factor 4x^2-9+2
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
I'm assuming that it's really 4x%5E2-9x%2B2

Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


Looking at the expression 4x%5E2-9x%2B2, we can see that the first coefficient is 4, the second coefficient is -9, and the last term is 2.



Now multiply the first coefficient 4 by the last term 2 to get %284%29%282%29=8.



Now the question is: what two whole numbers multiply to 8 (the previous product) and add to the second coefficient -9?



To find these two numbers, we need to list all of the factors of 8 (the previous product).



Factors of 8:

1,2,4,8

-1,-2,-4,-8



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



These factors pair up and multiply to 8.

1*8 = 8
2*4 = 8
(-1)*(-8) = 8
(-2)*(-4) = 8


Now let's add up each pair of factors to see if one pair adds to the middle coefficient -9:



First NumberSecond NumberSum
181+8=9
242+4=6
-1-8-1+(-8)=-9
-2-4-2+(-4)=-6




From the table, we can see that the two numbers -1 and -8 add to -9 (the middle coefficient).



So the two numbers -1 and -8 both multiply to 8 and add to -9



Now replace the middle term -9x with -x-8x. Remember, -1 and -8 add to -9. So this shows us that -x-8x=-9x.



4x%5E2%2Bhighlight%28-x-8x%29%2B2 Replace the second term -9x with -x-8x.



%284x%5E2-x%29%2B%28-8x%2B2%29 Group the terms into two pairs.



x%284x-1%29%2B%28-8x%2B2%29 Factor out the GCF x from the first group.



x%284x-1%29-2%284x-1%29 Factor out 2 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.



%28x-2%29%284x-1%29 Combine like terms. Or factor out the common term 4x-1



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Answer:



So 4%2Ax%5E2-9%2Ax%2B2 factors to %28x-2%29%284x-1%29.



In other words, 4%2Ax%5E2-9%2Ax%2B2=%28x-2%29%284x-1%29.



Note: you can check the answer by expanding %28x-2%29%284x-1%29 to get 4%2Ax%5E2-9%2Ax%2B2 or by graphing the original expression and the answer (the two graphs should be identical).