SOLUTION: 9x² + 49x + 20

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Question 882098: 9x² + 49x + 20
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
I'm assuming you want to factor.


Looking at the expression 9x%5E2%2B49x%2B20, we can see that the first coefficient is 9, the second coefficient is 49, and the last term is 20.


Now multiply the first coefficient 9 by the last term 20 to get %289%29%2820%29=180.


Now the question is: what two whole numbers multiply to 180 (the previous product) and add to the second coefficient 49?


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 49:


First NumberSecond NumberSum
11801+180=181
2902+90=92
3603+60=63
4454+45=49
5365+36=41
6306+30=36
9209+20=29
101810+18=28
121512+15=27
-1-180-1+(-180)=-181
-2-90-2+(-90)=-92
-3-60-3+(-60)=-63
-4-45-4+(-45)=-49
-5-36-5+(-36)=-41
-6-30-6+(-30)=-36
-9-20-9+(-20)=-29
-10-18-10+(-18)=-28
-12-15-12+(-15)=-27



From the table, we can see that the two numbers 4 and 45 add to 49 (the middle coefficient).


So the two numbers 4 and 45 both multiply to 180 and add to 49


Now replace the middle term 49x with 4x%2B45x. Remember, 4 and 45 add to 49. So this shows us that 4x%2B45x=49x.


9x%5E2%2Bhighlight%284x%2B45x%29%2B20 Replace the second term 49x with 4x%2B45x.


%289x%5E2%2B4x%29%2B%2845x%2B20%29 Group the terms into two pairs.


x%289x%2B4%29%2B%2845x%2B20%29 Factor out the GCF x from the first group.


x%289x%2B4%29%2B5%289x%2B4%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.


%28x%2B5%29%289x%2B4%29 Combine like terms. Or factor out the common term 9x%2B4


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Answer:


So 9x%5E2%2B49x%2B20 factors to %28x%2B5%29%289x%2B4%29.


In other words, 9x%5E2%2B49x%2B20=%28x%2B5%29%289x%2B4%29.


Note: you can check the answer by expanding %28x%2B5%29%289x%2B4%29 to get 9x%5E2%2B49x%2B20 or by graphing the original expression and the answer (the two graphs should be identical).