SOLUTION: How do I factor this? 4a^2-4a+1=

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Question 455196: How do I factor this?
4a^2-4a+1=

Answer by richwmiller(17219) About Me  (Show Source):
You can put this solution on YOUR website!
(2a-1)^2
or
(1-2a)^2
you want factors of 4 that add up to -4
Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


Looking at the expression 4a%5E2-4a%2B1, we can see that the first coefficient is 4, the second coefficient is -4, and the last term is 1.



Now multiply the first coefficient 4 by the last term 1 to get %284%29%281%29=4.



Now the question is: what two whole numbers multiply to 4 (the previous product) and add to the second coefficient -4?



To find these two numbers, we need to list all of the factors of 4 (the previous product).



Factors of 4:

1,2,4

-1,-2,-4



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



These factors pair up and multiply to 4.

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


Now let's add up each pair of factors to see if one pair adds to the middle coefficient -4:



First NumberSecond NumberSum
141+4=5
222+2=4
-1-4-1+(-4)=-5
-2-2-2+(-2)=-4




From the table, we can see that the two numbers -2 and -2 add to -4 (the middle coefficient).



So the two numbers -2 and -2 both multiply to 4 and add to -4



Now replace the middle term -4a with -2a-2a. Remember, -2 and -2 add to -4. So this shows us that -2a-2a=-4a.



4a%5E2%2Bhighlight%28-2a-2a%29%2B1 Replace the second term -4a with -2a-2a.



%284a%5E2-2a%29%2B%28-2a%2B1%29 Group the terms into two pairs.



2a%282a-1%29%2B%28-2a%2B1%29 Factor out the GCF 2a from the first group.



2a%282a-1%29-1%282a-1%29 Factor out 1 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.



%282a-1%29%282a-1%29 Combine like terms. Or factor out the common term 2a-1



%282a-1%29%5E2 Condense the terms.



===============================================================



Answer:



So 4%2Aa%5E2-4%2Aa%2B1 factors to %282a-1%29%5E2.



In other words, 4%2Aa%5E2-4%2Aa%2B1=%282a-1%29%5E2.



Note: you can check the answer by expanding %282a-1%29%5E2 to get 4%2Aa%5E2-4%2Aa%2B1 or by graphing the original expression and the answer (the two graphs should be identical).