SOLUTION: Factor 81x^2-126xy+49y^2

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Question 610312: Factor 81x^2-126xy+49y^2
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Looking at the expression 81x%5E2-126xy%2B49y%5E2, we can see that the first coefficient is 81, the second coefficient is -126, and the last coefficient is 49.


Now multiply the first coefficient 81 by the last coefficient 49 to get %2881%29%2849%29=3969.


Now the question is: what two whole numbers multiply to 3969 (the previous product) and add to the second coefficient -126?


To find these two numbers, we need to list all of the factors of 3969 (the previous product).


Factors of 3969:
1,3,7,9,21,27,49,63,81,147,189,441,567,1323,3969
-1,-3,-7,-9,-21,-27,-49,-63,-81,-147,-189,-441,-567,-1323,-3969


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


These factors pair up and multiply to 3969.
1*3969 = 3969
3*1323 = 3969
7*567 = 3969
9*441 = 3969
21*189 = 3969
27*147 = 3969
49*81 = 3969
63*63 = 3969
(-1)*(-3969) = 3969
(-3)*(-1323) = 3969
(-7)*(-567) = 3969
(-9)*(-441) = 3969
(-21)*(-189) = 3969
(-27)*(-147) = 3969
(-49)*(-81) = 3969
(-63)*(-63) = 3969

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -126:


First NumberSecond NumberSum
139691+3969=3970
313233+1323=1326
75677+567=574
94419+441=450
2118921+189=210
2714727+147=174
498149+81=130
636363+63=126
-1-3969-1+(-3969)=-3970
-3-1323-3+(-1323)=-1326
-7-567-7+(-567)=-574
-9-441-9+(-441)=-450
-21-189-21+(-189)=-210
-27-147-27+(-147)=-174
-49-81-49+(-81)=-130
-63-63-63+(-63)=-126



From the table, we can see that the two numbers -63 and -63 add to -126 (the middle coefficient).


So the two numbers -63 and -63 both multiply to 3969 and add to -126


Now replace the middle term -126xy with -63xy-63xy. Remember, -63 and -63 add to -126. So this shows us that -63xy-63xy=-126xy.


81x%5E2%2Bhighlight%28-63xy-63xy%29%2B49y%5E2 Replace the second term -126xy with -63xy-63xy.


%2881x%5E2-63xy%29%2B%28-63xy%2B49y%5E2%29 Group the terms into two pairs.


9x%289x-7y%29%2B%28-63xy%2B49y%5E2%29 Factor out the GCF 9x from the first group.


9x%289x-7y%29-7y%289x-7y%29 Factor out the-7y from the second group.


%289x-7y%29%289x-7y%29 Factor out the9x-7y from the entire expression.


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Answer:


So 81x%5E2-126xy%2B49y%5E2 completely factors to %289x-7y%29%289x-7y%29


In other words, 81x%5E2-126xy%2B49y%5E2=%289x-7y%29%289x-7y%29 for all values of x and y.


Note: you can check the answer by expanding %289x-7y%29%289x-7y%29 to get 81x%5E2-126xy%2B49y%5E2 back again or by graphing the original expression and the answer (the two graphs should be identical).