SOLUTION: 3d^2-17d+20

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Question 604156: 3d^2-17d+20
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
It looks like you want to factor this.




Looking at the expression 3d%5E2-17d%2B20, we can see that the first coefficient is 3, the second coefficient is -17, and the last term is 20.


Now multiply the first coefficient 3 by the last term 20 to get %283%29%2820%29=60.


Now the question is: what two whole numbers multiply to 60 (the previous product) and add to the second coefficient -17?


To find these two numbers, we need to list all of the factors of 60 (the previous product).


Factors of 60:
1,2,3,4,5,6,10,12,15,20,30,60
-1,-2,-3,-4,-5,-6,-10,-12,-15,-20,-30,-60


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


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

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -17:


First NumberSecond NumberSum
1601+60=61
2302+30=32
3203+20=23
4154+15=19
5125+12=17
6106+10=16
-1-60-1+(-60)=-61
-2-30-2+(-30)=-32
-3-20-3+(-20)=-23
-4-15-4+(-15)=-19
-5-12-5+(-12)=-17
-6-10-6+(-10)=-16



From the table, we can see that the two numbers -5 and -12 add to -17 (the middle coefficient).


So the two numbers -5 and -12 both multiply to 60 and add to -17


Now replace the middle term -17d with -5d-12d. Remember, -5 and -12 add to -17. So this shows us that -5d-12d=-17d.


3d%5E2%2Bhighlight%28-5d-12d%29%2B20 Replace the second term -17d with -5d-12d.


%283d%5E2-5d%29%2B%28-12d%2B20%29 Group the terms into two pairs.


d%283d-5%29%2B%28-12d%2B20%29 Factor out the GCF d from the first group.


d%283d-5%29-4%283d-5%29 Factor out 4 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.


%28d-4%29%283d-5%29 Combine like terms. Or factor out the common term 3d-5


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Answer:


So 3d%5E2-17d%2B20 factors to %28d-4%29%283d-5%29.


In other words, 3d%5E2-17d%2B20=%28d-4%29%283d-5%29.


Note: you can check the answer by expanding %28d-4%29%283d-5%29 to get 3d%5E2-17d%2B20 or by graphing the original expression and the answer (the two graphs should be identical).