SOLUTION: Factor the following expression completely. x^2 - 8x + 16 - 9y^2

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Question 318020: Factor the following expression completely.
x^2 - 8x + 16 - 9y^2
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
First let's focus on x%5E2+-+8x+%2B+16


Looking at the expression x%5E2-8x%2B16, we can see that the first coefficient is 1, the second coefficient is -8, and the last term is 16.


Now multiply the first coefficient 1 by the last term 16 to get %281%29%2816%29=16.


Now the question is: what two whole numbers multiply to 16 (the previous product) and add to the second coefficient -8?


To find these two numbers, we need to list all of the factors of 16 (the previous product).


Factors of 16:
1,2,4,8,16
-1,-2,-4,-8,-16


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


These factors pair up and multiply to 16.
1*16 = 16
2*8 = 16
4*4 = 16
(-1)*(-16) = 16
(-2)*(-8) = 16
(-4)*(-4) = 16

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -8:


First NumberSecond NumberSum
1161+16=17
282+8=10
444+4=8
-1-16-1+(-16)=-17
-2-8-2+(-8)=-10
-4-4-4+(-4)=-8



From the table, we can see that the two numbers -4 and -4 add to -8 (the middle coefficient).


So the two numbers -4 and -4 both multiply to 16 and add to -8


Now replace the middle term -8x with -4x-4x. Remember, -4 and -4 add to -8. So this shows us that -4x-4x=-8x.


x%5E2%2Bhighlight%28-4x-4x%29%2B16 Replace the second term -8x with -4x-4x.


%28x%5E2-4x%29%2B%28-4x%2B16%29 Group the terms into two pairs.


x%28x-4%29%2B%28-4x%2B16%29 Factor out the GCF x from the first group.


x%28x-4%29-4%28x-4%29 Factor out 4 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.


%28x-4%29%28x-4%29 Combine like terms. Or factor out the common term x-4


%28x-4%29%5E2 Condense the terms.


So x%5E2-8x%2B16 factors to %28x-4%29%5E2.


In other words, x%5E2-8x%2B16=%28x-4%29%5E2.


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So x%5E2+-+8x+%2B+16-9y%5E2=%28x-4%29%5E2-9y%5E2


Now let's factor %28x-4%29%5E2-9y%5E2



%28x-4%29%5E2-9y%5E2 Start with the given expression.


%28x-4%29%5E2-%283y%29%5E2 Rewrite 9y%5E2 as %283y%29%5E2.


Notice how we have a difference of squares A%5E2-B%5E2 where in this case A=x-4 and B=3y.


So let's use the difference of squares formula A%5E2-B%5E2=%28A%2BB%29%28A-B%29 to factor the expression:


A%5E2-B%5E2=%28A%2BB%29%28A-B%29 Start with the difference of squares formula.


%28x-4%29%5E2-%283y%29%5E2=%28x-4%2B3y%29%28x-4-3y%29 Plug in A=x-4 and B=3y.


So this shows us that %28x-4%29%5E2-9y%5E2 factors to %28x-4%2B3y%29%28x-4-3y%29.


So this then means that x%5E2+-+8x+%2B+16-9y%5E2=%28x-4%2B3y%29%28x-4-3y%29.