document.write( "Question 168159: 3x^2+37xy-86y^2 \n" ); document.write( "

Algebra.Com's Answer #123936 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at $\"3x%5E2%2B37xy-86y%5E2\"$ we can see that the first term is $\"3x%5E2\"$ and the last term is $\"-86y%5E2\"$ where the coefficients are 3 and -86 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 3 and the last coefficient -86 to get -258. Now what two numbers multiply to -258 and add to the middle coefficient 37? Let's list all of the factors of -258:\r
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\n" ); document.write( "\n" ); document.write( "Factors of -258:\r
\n" ); document.write( "\n" ); document.write( "1,2,3,6,43,86,129,258\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-6,-43,-86,-129,-258 ...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 -258\r
\n" ); document.write( "\n" ); document.write( "(1)*(-258)\r
\n" ); document.write( "\n" ); document.write( "(2)*(-129)\r
\n" ); document.write( "\n" ); document.write( "(3)*(-86)\r
\n" ); document.write( "\n" ); document.write( "(6)*(-43)\r
\n" ); document.write( "\n" ); document.write( "(-1)*(258)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(129)\r
\n" ); document.write( "\n" ); document.write( "(-3)*(86)\r
\n" ); document.write( "\n" ); document.write( "(-6)*(43)\r
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\n" ); document.write( "\n" ); document.write( "note: remember, the product of a negative and a positive number is a negative number\r
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\n" ); document.write( "\n" ); document.write( "Now which of these pairs add to 37? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 37\r
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First Number	Second Number	Sum
1	-258	1+(-258)=-257
2	-129	2+(-129)=-127
3	-86	3+(-86)=-83
6	-43	6+(-43)=-37
-1	258	-1+258=257
-2	129	-2+129=127
-3	86	-3+86=83
-6	43	-6+43=37

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\n" ); document.write( "\n" ); document.write( "From this list we can see that -6 and 43 add up to 37 and multiply to -258\r
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"3x%5E2%2B37xy-86y%5E2\"$ , replace $\"37xy\"$ with $\"-6xy%2B43xy\"$ (notice $\"-6xy%2B43xy\"$ adds up to $\"37xy\"$ . So it is equivalent to $\"37xy\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"3x%5E2%2Bhighlight%28-6xy%2B43xy%29%2B-86y%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"3x%5E2-6xy%2B43xy-86y%5E2\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%283x%5E2-6xy%29%2B%2843xy-86y%5E2%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"3x%28x-2y%29%2B43y%28x-2y%29\"$ Factor out the GCF of $\"3x\"$ out of the first group. Factor out the GCF of $\"43y\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%283x%2B43y%29%28x-2y%29\"$ Since we have a common term of $\"x-2y\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"3x%5E2-6xy%2B43xy-86y%5E2\"$ factors to $\"%283x%2B43y%29%28x-2y%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So this also means that $\"3x%5E2%2B37xy-86y%5E2\"$ factors to $\"%283x%2B43y%29%28x-2y%29\"$ (since $\"3x%5E2%2B37xy-86y%5E2\"$ is equivalent to $\"3x%5E2-6xy%2B43xy-86y%5E2\"$ )\r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"3x%5E2%2B37xy-86y%5E2\"$ factors to $\"%283x%2B43y%29%28x-2y%29\"$ \r
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