Question 118297: How can you factor out, completely the problem 4t^2-12t+9-w^2?
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website!  Start with the given expression
Let's focus on the polynomial 
Looking at we can see that the first term is  and the last term is  where the coefficients are 4 and 9 respectively.
Now multiply the first coefficient 4 and the last coefficient 9 to get 36. Now what two numbers multiply to 36 and add to the middle coefficient -12? Let's list all of the factors of 36:
Factors of 36:
1,2,3,4,6,9,12,18
-1,-2,-3,-4,-6,-9,-12,-18 ...List the negative factors as well. This will allow us to find all possible combinations
These factors pair up and multiply to 36
1*36
2*18
3*12
4*9
6*6
(-1)*(-36)
(-2)*(-18)
(-3)*(-12)
(-4)*(-9)
(-6)*(-6)
note: remember two negative numbers multiplied together make a positive number
Now which of these pairs add to -12? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -12
| First Number | Second Number | Sum | | 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 | | -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 this list we can see that -6 and -6 add up to -12 and multiply to 36
Now looking at the expression , replace  with  (notice adds up to . So it is equivalent to )
Now let's factor  by grouping:
 Group like terms
 Factor out the GCF of  out of the first group. Factor out the GCF of  out of the second group
 Since we have a common term of  , we can combine like terms
So factors to which is 
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So becomes 
Notice how we have a difference of squares. This means we can use the difference of squares formula to factor further.
Let  and 
 So we then get this
 Now factor using the difference of squares
 Now replace A with and B with 
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Answer:
So completely factors to
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