document.write( "Question 599427: How do you factor x^2 + 5xy - 14y^2 ? \n" ); document.write( "

Algebra.Com's Answer #378941 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"x%5E2%2B5xy-14y%5E2\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"5\"$ , and the last coefficient is $\"-14\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last coefficient $\"-14\"$ to get $\"%281%29%28-14%29=-14\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-14\"$ (the previous product) and add to the second coefficient $\"5\"$ ?\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 $\"-14\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-14\"$ :\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-7,-14\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 $\"-14\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-14) = -14
\n" ); document.write( "2*(-7) = -14
\n" ); document.write( "(-1)*(14) = -14
\n" ); document.write( "(-2)*(7) = -14\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 $\"5\"$ :\r
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First Number	Second Number	Sum
1	-14	1+(-14)=-13
2	-7	2+(-7)=-5
-1	14	-1+14=13
-2	7	-2+7=5

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-2\"$ and $\"7\"$ add to $\"5\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-2\"$ and $\"7\"$ both multiply to $\"-14\"$ and add to $\"5\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"5xy\"$ with $\"-2xy%2B7xy\"$ . Remember, $\"-2\"$ and $\"7\"$ add to $\"5\"$ . So this shows us that $\"-2xy%2B7xy=5xy\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"x%5E2%2Bhighlight%28-2xy%2B7xy%29-14y%5E2\"$ Replace the second term $\"5xy\"$ with $\"-2xy%2B7xy\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%28x%5E2-2xy%29%2B%287xy-14y%5E2%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x-2y%29%2B%287xy-14y%5E2%29\"$ Factor out the GCF $\"x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x-2y%29%2B7y%28x-2y%29\"$ 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( " $\"%28x%2B7y%29%28x-2y%29\"$ Combine like terms. Or factor out the common term $\"x-2y\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"x%5E2%2B5xy-14y%5E2\"$ factors to $\"%28x%2B7y%29%28x-2y%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"x%5E2%2B5xy-14y%5E2=%28x%2B7y%29%28x-2y%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%28x%2B7y%29%28x-2y%29\"$ to get $\"x%5E2%2B5xy-14y%5E2\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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