SOLUTION: Factor 4n^2 -28n+49

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Question 738916: Factor 4n^2 -28n+49
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Looking at the expression 4n%5E2-28n%2B49, we can see that the first coefficient is 4, the second coefficient is -28, and the last term is 49.


Now multiply the first coefficient 4 by the last term 49 to get %284%29%2849%29=196.


Now the question is: what two whole numbers multiply to 196 (the previous product) and add to the second coefficient -28?


To find these two numbers, we need to list all of the factors of 196 (the previous product).


Factors of 196:
1,2,4,7,14,28,49,98,196
-1,-2,-4,-7,-14,-28,-49,-98,-196


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


These factors pair up and multiply to 196.
1*196 = 196
2*98 = 196
4*49 = 196
7*28 = 196
14*14 = 196
(-1)*(-196) = 196
(-2)*(-98) = 196
(-4)*(-49) = 196
(-7)*(-28) = 196
(-14)*(-14) = 196

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -28:


First NumberSecond NumberSum
11961+196=197
2982+98=100
4494+49=53
7287+28=35
141414+14=28
-1-196-1+(-196)=-197
-2-98-2+(-98)=-100
-4-49-4+(-49)=-53
-7-28-7+(-28)=-35
-14-14-14+(-14)=-28



From the table, we can see that the two numbers -14 and -14 add to -28 (the middle coefficient).


So the two numbers -14 and -14 both multiply to 196 and add to -28


Now replace the middle term -28n with -14n-14n. Remember, -14 and -14 add to -28. So this shows us that -14n-14n=-28n.


4n%5E2%2Bhighlight%28-14n-14n%29%2B49 Replace the second term -28n with -14n-14n.


%284n%5E2-14n%29%2B%28-14n%2B49%29 Group the terms into two pairs.


2n%282n-7%29%2B%28-14n%2B49%29 Factor out the GCF 2n from the first group.


2n%282n-7%29-7%282n-7%29 Factor out 7 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.


%282n-7%29%282n-7%29 Combine like terms. Or factor out the common term 2n-7


%282n-7%29%5E2 Condense the terms.


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Answer:


So 4n%5E2-28n%2B49 factors to %282n-7%29%5E2.


In other words, 4n%5E2-28n%2B49=%282n-7%29%5E2.


Note: you can check the answer by expanding %282n-7%29%5E2 to get 4n%5E2-28n%2B49 or by graphing the original expression and the answer (the two graphs should be identical).