SOLUTION: How can you factor out, completely the problem 4t^2-12t+9-w^2?

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

--------------------

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

---------------------------------

Answer:

So completely factors to
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