SOLUTION: Factor Completely 3a^2+5ab-12b^2

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Question 551360: Factor Completely
3a^2+5ab-12b^2
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Looking at the expression 3a%5E2%2B5ab-12b%5E2, we can see that the first coefficient is 3, the second coefficient is 5, and the last coefficient is -12.


Now multiply the first coefficient 3 by the last coefficient -12 to get %283%29%28-12%29=-36.


Now the question is: what two whole numbers multiply to -36 (the previous product) and add to the second coefficient 5?


To find these two numbers, we need to list all of the factors of -36 (the previous product).


Factors of -36:
1,2,3,4,6,9,12,18,36
-1,-2,-3,-4,-6,-9,-12,-18,-36


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


These factors pair up and multiply to -36.
1*(-36) = -36
2*(-18) = -36
3*(-12) = -36
4*(-9) = -36
6*(-6) = -36
(-1)*(36) = -36
(-2)*(18) = -36
(-3)*(12) = -36
(-4)*(9) = -36
(-6)*(6) = -36

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 5:


First NumberSecond NumberSum
1-361+(-36)=-35
2-182+(-18)=-16
3-123+(-12)=-9
4-94+(-9)=-5
6-66+(-6)=0
-136-1+36=35
-218-2+18=16
-312-3+12=9
-49-4+9=5
-66-6+6=0



From the table, we can see that the two numbers -4 and 9 add to 5 (the middle coefficient).


So the two numbers -4 and 9 both multiply to -36 and add to 5


Now replace the middle term 5ab with -4ab%2B9ab. Remember, -4 and 9 add to 5. So this shows us that -4ab%2B9ab=5ab.


3a%5E2%2Bhighlight%28-4ab%2B9ab%29-12b%5E2 Replace the second term 5ab with -4ab%2B9ab.


%283a%5E2-4ab%29%2B%289ab-12b%5E2%29 Group the terms into two pairs.


a%283a-4b%29%2B%289ab-12b%5E2%29 Factor out the GCF a from the first group.


a%283a-4b%29%2B3b%283a-4b%29 Factor out 3b 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.


%28a%2B3b%29%283a-4b%29 Combine like terms. Or factor out the common term 3a-4b


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Answer:


So 3a%5E2%2B5ab-12b%5E2 factors to %28a%2B3b%29%283a-4b%29.


In other words, 3a%5E2%2B5ab-12b%5E2=%28a%2B3b%29%283a-4b%29.


Note: you can check the answer by expanding %28a%2B3b%29%283a-4b%29 to get 3a%5E2%2B5ab-12b%5E2 or by graphing the original expression and the answer (the two graphs should be identical).