SOLUTION: Factor completely: x^2 - 8x + 16.

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Question 113857: Factor completely: x^2 - 8x + 16.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


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.



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Answer:



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



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



Note: you can check the answer by expanding %28x-4%29%5E2 to get x%5E2-8%2Ax%2B16 or by graphing the original expression and the answer (the two graphs should be identical).