SOLUTION: Factor 80+36t+4t^2

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Question 190287: Factor
80+36t+4t^2
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
80%2B36t%2B4t%5E2 Start with the given expression.


4t%5E2%2B36t%2B80 Rearrange the terms.


4%28t%5E2%2B9t%2B20%29 Factor out the GCF 4


Now let's focus on the inner expression t%5E2%2B9t%2B20


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Looking at the expression t%5E2%2B9t%2B20, we can see that the first coefficient is 1, the second coefficient is 9, and the last term is 20.


Now multiply the first coefficient 1 by the last term 20 to get %281%29%2820%29=20.


Now the question is: what two whole numbers multiply to 20 (the previous product) and add to the second coefficient 9?


To find these two numbers, we need to list all of the factors of 20 (the previous product).


Factors of 20:
1,2,4,5,10,20
-1,-2,-4,-5,-10,-20


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


These factors pair up and multiply to 20.
1*20
2*10
4*5
(-1)*(-20)
(-2)*(-10)
(-4)*(-5)

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 9:


First NumberSecond NumberSum
1201+20=21
2102+10=12
454+5=9
-1-20-1+(-20)=-21
-2-10-2+(-10)=-12
-4-5-4+(-5)=-9



From the table, we can see that the two numbers 4 and 5 add to 9 (the middle coefficient).


So the two numbers 4 and 5 both multiply to 20 and add to 9


Now replace the middle term 9t with 4t%2B5t. Remember, 4 and 5 add to 9. So this shows us that 4t%2B5t=9t.


t%5E2%2Bhighlight%284t%2B5t%29%2B20 Replace the second term 9t with 4t%2B5t.


%28t%5E2%2B4t%29%2B%285t%2B20%29 Group the terms into two pairs.


t%28t%2B4%29%2B%285t%2B20%29 Factor out the GCF t from the first group.


t%28t%2B4%29%2B5%28t%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.


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



So t%5E2%2B9t%2B20 factors to %28t%2B5%29%28t%2B4%29.


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So our expression goes from 4%28t%5E2%2B9t%2B20%29 and factors further to 4%28t%2B5%29%28t%2B4%29


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Answer:


So 80%2B36t%2B4t%5E2 completely factors to 4%28t%2B5%29%28t%2B4%29