document.write( "Question 124554: 6x^2+5xy-21y^2 \n" ); document.write( "

Algebra.Com's Answer #91284 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
Do you want to factor this?\r
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\n" ); document.write( "\n" ); document.write( "Looking at $\"6x%5E2%2B5xy-21y%5E2\"$ we can see that the first term is $\"6x%5E2\"$ and the last term is $\"-21y%5E2\"$ where the coefficients are 6 and -21 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 6 and the last coefficient -21 to get -126. Now what two numbers multiply to -126 and add to the middle coefficient 5? Let's list all of the factors of -126:\r
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\n" ); document.write( "\n" ); document.write( "Factors of -126:\r
\n" ); document.write( "\n" ); document.write( "1,2,3,6,7,9,14,18,21,42,63,126\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-6,-7,-9,-14,-18,-21,-42,-63,-126 ...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 -126\r
\n" ); document.write( "\n" ); document.write( "(1)*(-126)\r
\n" ); document.write( "\n" ); document.write( "(2)*(-63)\r
\n" ); document.write( "\n" ); document.write( "(3)*(-42)\r
\n" ); document.write( "\n" ); document.write( "(6)*(-21)\r
\n" ); document.write( "\n" ); document.write( "(7)*(-18)\r
\n" ); document.write( "\n" ); document.write( "(9)*(-14)\r
\n" ); document.write( "\n" ); document.write( "(-1)*(126)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(63)\r
\n" ); document.write( "\n" ); document.write( "(-3)*(42)\r
\n" ); document.write( "\n" ); document.write( "(-6)*(21)\r
\n" ); document.write( "\n" ); document.write( "(-7)*(18)\r
\n" ); document.write( "\n" ); document.write( "(-9)*(14)\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 5? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 5\r
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First Number	Second Number	Sum
1	-126	1+(-126)=-125
2	-63	2+(-63)=-61
3	-42	3+(-42)=-39
6	-21	6+(-21)=-15
7	-18	7+(-18)=-11
9	-14	9+(-14)=-5
-1	126	-1+126=125
-2	63	-2+63=61
-3	42	-3+42=39
-6	21	-6+21=15
-7	18	-7+18=11
-9	14	-9+14=5

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\n" ); document.write( "\n" ); document.write( "From this list we can see that -9 and 14 add up to 5 and multiply to -126\r
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"6x%5E2%2B5xy-21y%5E2\"$ , replace $\"5xy\"$ with $\"-9xy%2B14xy\"$ (notice $\"-9xy%2B14xy\"$ adds up to $\"5xy\"$ . So it is equivalent to $\"5xy\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"6x%5E2%2Bhighlight%28-9xy%2B14xy%29%2B-21y%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"6x%5E2-9xy%2B14xy-21y%5E2\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%286x%5E2-9xy%29%2B%2814xy-21y%5E2%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"3x%282x-3y%29%2B7y%282x-3y%29\"$ Factor out the GCF of $\"3x\"$ out of the first group. Factor out the GCF of $\"7y\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%283x%2B7y%29%282x-3y%29\"$ Since we have a common term of $\"2x-3y\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"6x%5E2-9xy%2B14xy-21y%5E2\"$ factors to $\"%283x%2B7y%29%282x-3y%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So this also means that $\"6x%5E2%2B5xy-21y%5E2\"$ factors to $\"%283x%2B7y%29%282x-3y%29\"$ (since $\"6x%5E2%2B5xy-21y%5E2\"$ is equivalent to $\"6x%5E2-9xy%2B14xy-21y%5E2\"$ )\r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"6x%5E2%2B5xy-21y%5E2\"$ factors to $\"%283x%2B7y%29%282x-3y%29\"$ \n" ); document.write( "

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