document.write( "Question 127254: Factor x^2+(x-y)^2-2x*(x-y) Thank you, this question is not from text book \n" ); document.write( "

Algebra.Com's Answer #93221 by jim_thompson5910(35256) $\"\"$ $\"About$

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$\"x%5E2%2B%28x-y%29%5E2-2x%2A%28x-y%29\"$ Start with the given expression\r
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\n" ); document.write( "\n" ); document.write( " $\"%28x-y%29%5E2-2x%2A%28x-y%29%2Bx%5E2\"$ Rearrange the terms\r
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\n" ); document.write( "\n" ); document.write( "Now let $\"z=x-y\"$ . So our equation becomes \r
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\n" ); document.write( "\n" ); document.write( " $\"z%5E2-2xz%2Bx%5E2\"$ Notice how I replaced each $\"x-y\"$ with z\r
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\n" ); document.write( "\n" ); document.write( "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.\r
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\n" ); document.write( "\n" ); document.write( "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:\r
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\n" ); document.write( "\n" ); document.write( "Factors of 1:\r
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\n" ); document.write( "\n" ); document.write( "-1 ...List the negative factors as well. This will allow us to find all possible combinations\r
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\n" ); document.write( "\n" ); document.write( "These factors pair up and multiply to 1\r
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\n" ); document.write( "\n" ); document.write( "note: remember two negative numbers multiplied together make a positive number\r
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\n" ); document.write( "\n" ); document.write( "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\r
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First Number	Second Number	Sum
1	1	1+1=2
-1	-1	-1+(-1)=-2

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\n" ); document.write( "\n" ); document.write( "From this list we can see that -1 and -1 add up to -2 and multiply to 1\r
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\n" ); document.write( "\n" ); document.write( "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 $\"-2xz\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"1z%5E2%2Bhighlight%28-1xz%2B-1xz%29%2B1x%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"1z%5E2-1xz-1xz%2B1x%5E2\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%281z%5E2-1xz%29%2B%28-1xz%2B1x%5E2%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"z%28z-x%29-x%28z-x%29\"$ Factor out the GCF of $\"z\"$ out of the first group. Factor out the GCF of $\"-x\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%28z-x%29%28z-x%29\"$ Since we have a common term of $\"z-x\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"1z%5E2-1xz-1xz%2B1x%5E2\"$ factors to $\"%28z-x%29%28z-x%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "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 $\"1z%5E2-1xz-1xz%2B1x%5E2\"$ )\r
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\n" ); document.write( "\n" ); document.write( "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 $\"%28z-x%29%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"z%5E2-2xz%2B1x%5E2\"$ factors to $\"%28z-x%29%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"%28x-y-x%29%5E2\"$ Remember, we let $\"z=x-y\"$ . So replace each z with $\"x-y\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"%28-y%29%5E2\"$ Combine like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"y%5E2\"$ Square -y to get $\"y%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer: \r
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\n" ); document.write( "\n" ); document.write( "So $\"x%5E2%2B%28x-y%29%5E2-2x%2A%28x-y%29\"$ factors to $\"y%5E2\"$ \n" ); document.write( "

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