SOLUTION: Factor the quadratic expression x^2-2x+1

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Question 297233: Factor the quadratic expression
x^2-2x+1
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%5E2-2x%2B1, we can see that the first coefficient is 1, the second coefficient is -2, and the last term is 1.



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



Now the question is: what two whole numbers multiply to 1 (the previous product) and add to the second coefficient -2?



To find these two numbers, we need to list all of the factors of 1 (the previous product).



Factors of 1:

1

-1



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



These factors pair up and multiply to 1.

1*1 = 1
(-1)*(-1) = 1


Now let's add up each pair of factors to see if one pair adds to the middle coefficient -2:



First NumberSecond NumberSum
111+1=2
-1-1-1+(-1)=-2




From the table, we can see that the two numbers -1 and -1 add to -2 (the middle coefficient).



So the two numbers -1 and -1 both multiply to 1 and add to -2



Now replace the middle term -2x with -x-x. Remember, -1 and -1 add to -2. So this shows us that -x-x=-2x.



x%5E2%2Bhighlight%28-x-x%29%2B1 Replace the second term -2x with -x-x.



%28x%5E2-x%29%2B%28-x%2B1%29 Group the terms into two pairs.



x%28x-1%29%2B%28-x%2B1%29 Factor out the GCF x from the first group.



x%28x-1%29-1%28x-1%29 Factor out 1 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-1%29%28x-1%29 Combine like terms. Or factor out the common term x-1



%28x-1%29%5E2 Condense the terms.



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Answer:



So x%5E2-2%2Ax%2B1 factors to %28x-1%29%5E2.



In other words, x%5E2-2%2Ax%2B1=%28x-1%29%5E2.



Note: you can check the answer by expanding %28x-1%29%5E2 to get x%5E2-2%2Ax%2B1 or by graphing the original expression and the answer (the two graphs should be identical).