SOLUTION: In class we are trying to factor special products; trinomials to be specific. We need to get it in perfect square trinomial pattern. Here's the question: 9t^2-12t+4 Thanks

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: In class we are trying to factor special products; trinomials to be specific. We need to get it in perfect square trinomial pattern. Here's the question: 9t^2-12t+4 Thanks      Log On


   

Question 387220: In class we are trying to factor special products; trinomials to be specific. We need to get it in perfect square trinomial pattern. Here's the question:
9t^2-12t+4
Thanks
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Looking at the expression 9t%5E2-12t%2B4, we can see that the first coefficient is 9, the second coefficient is -12, and the last term is 4.


Now multiply the first coefficient 9 by the last term 4 to get %289%29%284%29=36.


Now the question is: what two whole numbers multiply to 36 (the previous product) and add to the second coefficient -12?


To find these two numbers, we need to list all of the factors of 36 (the previous product).


Factors of 36:
1,2,3,4,6,9,12,18,36
-1,-2,-3,-4,-6,-9,-12,-18,-36


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


These factors pair up and multiply to 36.
1*36 = 36
2*18 = 36
3*12 = 36
4*9 = 36
6*6 = 36
(-1)*(-36) = 36
(-2)*(-18) = 36
(-3)*(-12) = 36
(-4)*(-9) = 36
(-6)*(-6) = 36

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -12:


First NumberSecond NumberSum
1361+36=37
2182+18=20
3123+12=15
494+9=13
666+6=12
-1-36-1+(-36)=-37
-2-18-2+(-18)=-20
-3-12-3+(-12)=-15
-4-9-4+(-9)=-13
-6-6-6+(-6)=-12



From the table, we can see that the two numbers -6 and -6 add to -12 (the middle coefficient).


So the two numbers -6 and -6 both multiply to 36 and add to -12


Now replace the middle term -12t with -6t-6t. Remember, -6 and -6 add to -12. So this shows us that -6t-6t=-12t.


9t%5E2%2Bhighlight%28-6t-6t%29%2B4 Replace the second term -12t with -6t-6t.


%289t%5E2-6t%29%2B%28-6t%2B4%29 Group the terms into two pairs.


3t%283t-2%29%2B%28-6t%2B4%29 Factor out the GCF 3t from the first group.


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


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


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Answer:


So 9t%5E2-12t%2B4 factors to %283t-2%29%5E2.


In other words, 9t%5E2-12t%2B4=%283t-2%29%5E2.


Note: you can check the answer by expanding %283t-2%29%5E2 to get 9t%5E2-12t%2B4 or by graphing the original expression and the answer (the two graphs should be identical).


If you need more help, email me at jim_thompson5910@hotmail.com

Also, feel free to check out my tutoring website

Jim