SOLUTION: reverse foil or trial factoring 9x^2-25

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Question 299484: reverse foil or trial factoring
9x^2-25
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Note: 9x%5E2-25 can be written as 9x%5E2%2B0x-25


Looking at the expression 9x%5E2%2B0x-25, we can see that the first coefficient is 9, the second coefficient is 0, 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 0?


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 0:


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 -15 and 15 add to 0 (the middle coefficient).


So the two numbers -15 and 15 both multiply to -225 and add to 0


Now replace the middle term 0x with -15x%2B15x. Remember, -15 and 15 add to 0. So this shows us that -15x%2B15x=0x.


9x%5E2%2Bhighlight%28-15x%2B15x%29-25 Replace the second term 0x with -15x%2B15x.


%289x%5E2-15x%29%2B%2815x-25%29 Group the terms into two pairs.


3x%283x-5%29%2B%2815x-25%29 Factor out the GCF 3x from the first group.


3x%283x-5%29%2B5%283x-5%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.


%283x%2B5%29%283x-5%29 Combine like terms. Or factor out the common term 3x-5


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Answer:


So 9x%5E2%2B0x-25 factors to %283x%2B5%29%283x-5%29.


In other words, 9x%5E2%2B0x-25=%283x%2B5%29%283x-5%29.


So 9x%5E2-25=%283x%2B5%29%283x-5%29.


Note: you can check the answer by expanding %283x%2B5%29%283x-5%29 to get 9x%5E2%2B0x-25 or by graphing the original expression and the answer (the two graphs should be identical). Also, 9x%5E2-25 is a difference of squares.