SOLUTION: I am having difficulty factoring a problem. 49m^2-42m+9 Any help would be appreciated

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Question 634761: I am having difficulty factoring a problem. 49m^2-42m+9 Any help would be appreciated
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Looking at the expression 49m%5E2-42m%2B9, we can see that the first coefficient is 49, the second coefficient is -42, and the last term is 9.


Now multiply the first coefficient 49 by the last term 9 to get %2849%29%289%29=441.


Now the question is: what two whole numbers multiply to 441 (the previous product) and add to the second coefficient -42?


To find these two numbers, we need to list all of the factors of 441 (the previous product).


Factors of 441:
1,3,7,9,21,49,63,147,441
-1,-3,-7,-9,-21,-49,-63,-147,-441


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


These factors pair up and multiply to 441.
1*441 = 441
3*147 = 441
7*63 = 441
9*49 = 441
21*21 = 441
(-1)*(-441) = 441
(-3)*(-147) = 441
(-7)*(-63) = 441
(-9)*(-49) = 441
(-21)*(-21) = 441

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -42:


First NumberSecond NumberSum
14411+441=442
31473+147=150
7637+63=70
9499+49=58
212121+21=42
-1-441-1+(-441)=-442
-3-147-3+(-147)=-150
-7-63-7+(-63)=-70
-9-49-9+(-49)=-58
-21-21-21+(-21)=-42



From the table, we can see that the two numbers -21 and -21 add to -42 (the middle coefficient).


So the two numbers -21 and -21 both multiply to 441 and add to -42


Now replace the middle term -42m with -21m-21m. Remember, -21 and -21 add to -42. So this shows us that -21m-21m=-42m.


49m%5E2%2Bhighlight%28-21m-21m%29%2B9 Replace the second term -42m with -21m-21m.


%2849m%5E2-21m%29%2B%28-21m%2B9%29 Group the terms into two pairs.


7m%287m-3%29%2B%28-21m%2B9%29 Factor out the GCF 7m from the first group.


7m%287m-3%29-3%287m-3%29 Factor out 3 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.


%287m-3%29%287m-3%29 Combine like terms. Or factor out the common term 7m-3


%287m-3%29%5E2 Condense the terms.


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Answer:


So 49m%5E2-42m%2B9 factors to %287m-3%29%5E2.


In other words, 49m%5E2-42m%2B9=%287m-3%29%5E2.


Note: you can check the answer by expanding %287m-3%29%5E2 to get 49m%5E2-42m%2B9 or by graphing the original expression and the answer (the two graphs should be identical).

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