SOLUTION: factor the trinomial 9a*2+3a-20

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Question 683092: factor the trinomial 9a*2+3a-20
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Looking at the expression 9a%5E2%2B3a-20, we can see that the first coefficient is 9, the second coefficient is 3, and the last term is -20.


Now multiply the first coefficient 9 by the last term -20 to get %289%29%28-20%29=-180.


Now the question is: what two whole numbers multiply to -180 (the previous product) and add to the second coefficient 3?


To find these two numbers, we need to list all of the factors of -180 (the previous product).


Factors of -180:
1,2,3,4,5,6,9,10,12,15,18,20,30,36,45,60,90,180
-1,-2,-3,-4,-5,-6,-9,-10,-12,-15,-18,-20,-30,-36,-45,-60,-90,-180


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


These factors pair up and multiply to -180.
1*(-180) = -180
2*(-90) = -180
3*(-60) = -180
4*(-45) = -180
5*(-36) = -180
6*(-30) = -180
9*(-20) = -180
10*(-18) = -180
12*(-15) = -180
(-1)*(180) = -180
(-2)*(90) = -180
(-3)*(60) = -180
(-4)*(45) = -180
(-5)*(36) = -180
(-6)*(30) = -180
(-9)*(20) = -180
(-10)*(18) = -180
(-12)*(15) = -180

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 3:


First NumberSecond NumberSum
1-1801+(-180)=-179
2-902+(-90)=-88
3-603+(-60)=-57
4-454+(-45)=-41
5-365+(-36)=-31
6-306+(-30)=-24
9-209+(-20)=-11
10-1810+(-18)=-8
12-1512+(-15)=-3
-1180-1+180=179
-290-2+90=88
-360-3+60=57
-445-4+45=41
-536-5+36=31
-630-6+30=24
-920-9+20=11
-1018-10+18=8
-1215-12+15=3



From the table, we can see that the two numbers -12 and 15 add to 3 (the middle coefficient).


So the two numbers -12 and 15 both multiply to -180 and add to 3


Now replace the middle term 3a with -12a%2B15a. Remember, -12 and 15 add to 3. So this shows us that -12a%2B15a=3a.


9a%5E2%2Bhighlight%28-12a%2B15a%29-20 Replace the second term 3a with -12a%2B15a.


%289a%5E2-12a%29%2B%2815a-20%29 Group the terms into two pairs.


3a%283a-4%29%2B%2815a-20%29 Factor out the GCF 3a from the first group.


3a%283a-4%29%2B5%283a-4%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.


%283a%2B5%29%283a-4%29 Combine like terms. Or factor out the common term 3a-4


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Answer:


So 9a%5E2%2B3a-20 factors to %283a%2B5%29%283a-4%29.


In other words, 9a%5E2%2B3a-20=%283a%2B5%29%283a-4%29.


Note: you can check the answer by expanding %283a%2B5%29%283a-4%29 to get 9a%5E2%2B3a-20 or by graphing the original expression and the answer (the two graphs should be identical).