SOLUTION: y^2+yt-12t^2=

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Question 626283: y^2+yt-12t^2=
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
I'm assuming you want to factor this.

Looking at the expression y%5E2%2Bty-12t%5E2, we can see that the first coefficient is 1, the second coefficient is 1, and the last coefficient is -12.


Now multiply the first coefficient 1 by the last coefficient -12 to get %281%29%28-12%29=-12.


Now the question is: what two whole numbers multiply to -12 (the previous product) and add to the second coefficient 1?


To find these two numbers, we need to list all of the factors of -12 (the previous product).


Factors of -12:
1,2,3,4,6,12
-1,-2,-3,-4,-6,-12


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


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

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 1:


First NumberSecond NumberSum
1-121+(-12)=-11
2-62+(-6)=-4
3-43+(-4)=-1
-112-1+12=11
-26-2+6=4
-34-3+4=1



From the table, we can see that the two numbers -3 and 4 add to 1 (the middle coefficient).


So the two numbers -3 and 4 both multiply to -12 and add to 1


Now replace the middle term 1ty with -3ty%2B4ty. Remember, -3 and 4 add to 1. So this shows us that -3ty%2B4ty=1ty.


y%5E2%2Bhighlight%28-3ty%2B4ty%29-12t%5E2 Replace the second term 1ty with -3ty%2B4ty.


%28y%5E2-3ty%29%2B%284ty-12t%5E2%29 Group the terms into two pairs.


y%28y-3t%29%2B%284ty-12t%5E2%29 Factor out the GCF y from the first group.


y%28y-3t%29%2B4t%28y-3t%29 Factor out 4t 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.


%28y%2B4t%29%28y-3t%29 Combine like terms. Or factor out the common term y-3t


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Answer:


So y%5E2%2Byt-12t%5E2 factors to %28y%2B4t%29%28y-3t%29.


In other words, y%5E2%2Byt-12t%5E2=%28y%2B4t%29%28y-3t%29.


Note: you can check the answer by expanding %28y%2B4t%29%28y-3t%29 to get y%5E2%2Byt-12t%5E2 or by graphing the original expression and the answer (the two graphs should be identical).

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