SOLUTION: 9t^2+30t+25

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Question 289215: 9t^2+30t+25
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
I'm assuming that you want to factor.



Looking at the expression 9t%5E2%2B30t%2B25, we can see that the first coefficient is 9, the second coefficient is 30, and the last term is 25.


Now multiply the first coefficient 9 by the last term 25 to get %289%29%2825%29=225.


Now the question is: what two whole numbers multiply to 225 (the previous product) and add to the second coefficient 30?


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 30:


First NumberSecond NumberSum
12251+225=226
3753+75=78
5455+45=50
9259+25=34
151515+15=30
-1-225-1+(-225)=-226
-3-75-3+(-75)=-78
-5-45-5+(-45)=-50
-9-25-9+(-25)=-34
-15-15-15+(-15)=-30



From the table, we can see that the two numbers 15 and 15 add to 30 (the middle coefficient).


So the two numbers 15 and 15 both multiply to 225 and add to 30


Now replace the middle term 30t with 15t%2B15t. Remember, 15 and 15 add to 30. So this shows us that 15t%2B15t=30t.


9t%5E2%2Bhighlight%2815t%2B15t%29%2B25 Replace the second term 30t with 15t%2B15t.


%289t%5E2%2B15t%29%2B%2815t%2B25%29 Group the terms into two pairs.


3t%283t%2B5%29%2B%2815t%2B25%29 Factor out the GCF 3t from the first group.


3t%283t%2B5%29%2B5%283t%2B5%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.


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


%283t%2B5%29%5E2 Condense the terms.


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Answer:


So 9t%5E2%2B30t%2B25 factors to %283t%2B5%29%5E2.


In other words, 9t%5E2%2B30t%2B25=%283t%2B5%29%5E2.


Note: you can check the answer by expanding %283t%2B5%29%5E2 to get 9t%5E2%2B30t%2B25 or by graphing the original expression and the answer (the two graphs should be identical).