SOLUTION: Factor x^2+(x-y)^2-2x*(x-y) Thank you, this question is not from text book

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Question 127254: Factor x^2+(x-y)^2-2x*(x-y) Thank you, this question is not from text book
Found 2 solutions by jim_thompson5910, MathLover1:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
x%5E2%2B%28x-y%29%5E2-2x%2A%28x-y%29 Start with the given expression


%28x-y%29%5E2-2x%2A%28x-y%29%2Bx%5E2 Rearrange the terms


Now let z=x-y. So our equation becomes


z%5E2-2xz%2Bx%5E2 Notice how I replaced each x-y with z


Looking at 1z%5E2-2xz%2B1x%5E2 we can see that the first term is 1z%5E2 and the last term is 1x%5E2 where the coefficients are 1 and 1 respectively.

Now multiply the first coefficient 1 and the last coefficient 1 to get 1. Now what two numbers multiply to 1 and add to the middle coefficient -2? Let's list all of the factors of 1:



Factors of 1:
1

-1 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to 1
1*1
(-1)*(-1)

note: remember two negative numbers multiplied together make a positive number


Now which of these pairs add to -2? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -2

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



From this list we can see that -1 and -1 add up to -2 and multiply to 1


Now looking at the expression 1z%5E2-2xz%2B1x%5E2, replace -2xz with -1xz%2B-1xz (notice -1xz%2B-1xz adds up to -2xz. So it is equivalent to -2xz)

1z%5E2%2Bhighlight%28-1xz%2B-1xz%29%2B1x%5E2


Now let's factor 1z%5E2-1xz-1xz%2B1x%5E2 by grouping:


%281z%5E2-1xz%29%2B%28-1xz%2B1x%5E2%29 Group like terms


z%28z-x%29-x%28z-x%29 Factor out the GCF of z out of the first group. Factor out the GCF of -x out of the second group


%28z-x%29%28z-x%29 Since we have a common term of z-x, we can combine like terms

So 1z%5E2-1xz-1xz%2B1x%5E2 factors to %28z-x%29%28z-x%29


So this also means that 1z%5E2-2xz%2B1x%5E2 factors to %28z-x%29%28z-x%29 (since 1z%5E2-2xz%2B1x%5E2 is equivalent to 1z%5E2-1xz-1xz%2B1x%5E2)


note: %28z-x%29%28z-x%29 is equivalent to %28z-x%29%5E2 since the term z-x occurs twice. So 1z%5E2-2xz%2B1x%5E2 also factors to %28z-x%29%5E2



So z%5E2-2xz%2B1x%5E2 factors to %28z-x%29%5E2



%28x-y-x%29%5E2 Remember, we let z=x-y. So replace each z with x-y


%28-y%29%5E2 Combine like terms


y%5E2 Square -y to get y%5E2



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Answer:


So x%5E2%2B%28x-y%29%5E2-2x%2A%28x-y%29 factors to y%5E2

Answer by MathLover1(20850) About Me  (Show Source):