SOLUTION: factor by grouping 9a^(2)-40a-25=

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Question 614551: factor by grouping
9a^(2)-40a-25=
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Looking at the expression 9a%5E2-40a-25, we can see that the first coefficient is 9, the second coefficient is -40, and the last term is -25.


Now multiply the first coefficient 9 by the last term -25 to get %289%29%28-25%29=-225.


Now the question is: what two whole numbers multiply to -225 (the previous product) and add to the second coefficient -40?


To find these two numbers, we need to list all of the factors of -225 (the previous product).


Factors of -225:
1,3,5,9,15,25,45,75,225
-1,-3,-5,-9,-15,-25,-45,-75,-225


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


These factors pair up and multiply to -225.
1*(-225) = -225
3*(-75) = -225
5*(-45) = -225
9*(-25) = -225
15*(-15) = -225
(-1)*(225) = -225
(-3)*(75) = -225
(-5)*(45) = -225
(-9)*(25) = -225
(-15)*(15) = -225

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -40:


First NumberSecond NumberSum
1-2251+(-225)=-224
3-753+(-75)=-72
5-455+(-45)=-40
9-259+(-25)=-16
15-1515+(-15)=0
-1225-1+225=224
-375-3+75=72
-545-5+45=40
-925-9+25=16
-1515-15+15=0



From the table, we can see that the two numbers 5 and -45 add to -40 (the middle coefficient).


So the two numbers 5 and -45 both multiply to -225 and add to -40


Now replace the middle term -40a with 5a-45a. Remember, 5 and -45 add to -40. So this shows us that 5a-45a=-40a.


9a%5E2%2Bhighlight%285a-45a%29-25 Replace the second term -40a with 5a-45a.


%289a%5E2%2B5a%29%2B%28-45a-25%29 Group the terms into two pairs.


a%289a%2B5%29%2B%28-45a-25%29 Factor out the GCF a from the first group.


a%289a%2B5%29-5%289a%2B5%29 Factor out 5 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-5%29%289a%2B5%29 Combine like terms. Or factor out the common term 9a%2B5


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Answer:


So 9a%5E2-40a-25 factors to %28a-5%29%289a%2B5%29.


In other words, 9a%5E2-40a-25=%28a-5%29%289a%2B5%29.


Note: you can check the answer by expanding %28a-5%29%289a%2B5%29 to get 9a%5E2-40a-25 or by graphing the original expression and the answer (the two graphs should be identical).