SOLUTION: Factor Completely. 63b^2+294bf+343f^2

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Question 581992: Factor Completely.
63b^2+294bf+343f^2
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

63b%5E2%2B294bf%2B343f%5E2 Start with the given expression.


7%289b%5E2%2B42bf%2B49f%5E2%29 Factor out the GCF 7.


Now let's try to factor the inner expression 9b%5E2%2B42bf%2B49f%5E2


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Looking at the expression 9b%5E2%2B42bf%2B49f%5E2, we can see that the first coefficient is 9, the second coefficient is 42, and the last coefficient is 49.


Now multiply the first coefficient 9 by the last coefficient 49 to get %289%29%2849%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 42bf with 21bf%2B21bf. Remember, 21 and 21 add to 42. So this shows us that 21bf%2B21bf=42bf.


9b%5E2%2Bhighlight%2821bf%2B21bf%29%2B49f%5E2 Replace the second term 42bf with 21bf%2B21bf.


%289b%5E2%2B21bf%29%2B%2821bf%2B49f%5E2%29 Group the terms into two pairs.


3b%283b%2B7f%29%2B%2821bf%2B49f%5E2%29 Factor out the GCF 3b from the first group.


3b%283b%2B7f%29%2B7f%283b%2B7f%29 Factor out 7f 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.


%283b%2B7f%29%283b%2B7f%29 Combine like terms. Or factor out the common term 3b%2B7f


%283b%2B7f%29%5E2 Condense the terms.


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So 7%289b%5E2%2B42bf%2B49f%5E2%29 then factors further to 7%283b%2B7f%29%5E2


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Answer:


So 63b%5E2%2B294bf%2B343f%5E2 completely factors to 7%283b%2B7f%29%5E2.


In other words, 63b%5E2%2B294bf%2B343f%5E2=7%283b%2B7f%29%5E2.


Note: you can check the answer by expanding 7%283b%2B7f%29%5E2 to get 63b%5E2%2B294bf%2B343f%5E2 or by graphing the original expression and the answer (the two graphs should be identical).