document.write( "Question 207723This question is from textbook
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\n" ); document.write( "9x^2+18xy-7y^2 \n" ); document.write( "

Algebra.Com's Answer #157111 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"9x%5E2%2B18xy-7y%5E2\"$ , we can see that the first coefficient is $\"9\"$ , the second coefficient is $\"18\"$ , and the last coefficient is $\"-7\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"9\"$ by the last coefficient $\"-7\"$ to get $\"%289%29%28-7%29=-63\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-63\"$ (the previous product) and add to the second coefficient $\"18\"$ ?\r
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\n" ); document.write( "\n" ); document.write( "To find these two numbers, we need to list all of the factors of $\"-63\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-63\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,3,7,9,21,63\r
\n" ); document.write( "\n" ); document.write( "-1,-3,-7,-9,-21,-63\r
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\n" ); document.write( "\n" ); document.write( "Note: list the negative of each factor. 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 $\"-63\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-63) = -63
\n" ); document.write( "3*(-21) = -63
\n" ); document.write( "7*(-9) = -63
\n" ); document.write( "(-1)*(63) = -63
\n" ); document.write( "(-3)*(21) = -63
\n" ); document.write( "(-7)*(9) = -63\r
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\n" ); document.write( "\n" ); document.write( "Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"18\"$ :\r
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First Number	Second Number	Sum
1	-63	1+(-63)=-62
3	-21	3+(-21)=-18
7	-9	7+(-9)=-2
-1	63	-1+63=62
-3	21	-3+21=18
-7	9	-7+9=2

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-3\"$ and $\"21\"$ add to $\"18\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-3\"$ and $\"21\"$ both multiply to $\"-63\"$ and add to $\"18\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"18xy\"$ with $\"-3xy%2B21xy\"$ . Remember, $\"-3\"$ and $\"21\"$ add to $\"18\"$ . So this shows us that $\"-3xy%2B21xy=18xy\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"9x%5E2%2Bhighlight%28-3xy%2B21xy%29-7y%5E2\"$ Replace the second term $\"18xy\"$ with $\"-3xy%2B21xy\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%289x%5E2-3xy%29%2B%2821xy-7y%5E2%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"3x%283x-y%29%2B%2821xy-7y%5E2%29\"$ Factor out the GCF $\"3x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"3x%283x-y%29%2B7y%283x-y%29\"$ Factor out $\"7y\"$ 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.\r
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\n" ); document.write( "\n" ); document.write( " $\"%283x%2B7y%29%283x-y%29\"$ Combine like terms. Or factor out the common term $\"3x-y\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"9x%5E2%2B18xy-7y%5E2\"$ factors to $\"%283x%2B7y%29%283x-y%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"9x%5E2%2B18xy-7y%5E2=%283x%2B7y%29%283x-y%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%283x%2B7y%29%283x-y%29\"$ to get $\"9x%5E2%2B18xy-7y%5E2\"$ or by graphing the original expression and the answer (the two graphs should be identical).\r
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