document.write( "Question 176941: Hello wonderful ones. I have yet another problem.
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\n" ); document.write( "\n" ); document.write( "3x^2+22xy+24y^2
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Algebra.Com's Answer #131986 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( "Looking at $\"3x%5E2%2B22xy%2B24y%5E2\"$ we can see that the first term is $\"3x%5E2\"$ and the last term is $\"24y%5E2\"$ where the coefficients are 3 and 24 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 3 and the last coefficient 24 to get 72. Now what two numbers multiply to 72 and add to the middle coefficient 22? Let's list all of the factors of 72:\r
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\n" ); document.write( "\n" ); document.write( "Factors of 72:\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-8,-9,-12,-18,-24,-36,-72 ...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 72\r
\n" ); document.write( "\n" ); document.write( "1*72\r
\n" ); document.write( "\n" ); document.write( "2*36\r
\n" ); document.write( "\n" ); document.write( "3*24\r
\n" ); document.write( "\n" ); document.write( "4*18\r
\n" ); document.write( "\n" ); document.write( "6*12\r
\n" ); document.write( "\n" ); document.write( "8*9\r
\n" ); document.write( "\n" ); document.write( "(-1)*(-72)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(-36)\r
\n" ); document.write( "\n" ); document.write( "(-3)*(-24)\r
\n" ); document.write( "\n" ); document.write( "(-4)*(-18)\r
\n" ); document.write( "\n" ); document.write( "(-6)*(-12)\r
\n" ); document.write( "\n" ); document.write( "(-8)*(-9)\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 22? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 22\r
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First Number	Second Number	Sum
1	72	1+72=73
2	36	2+36=38
3	24	3+24=27
4	18	4+18=22
6	12	6+12=18
8	9	8+9=17
-1	-72	-1+(-72)=-73
-2	-36	-2+(-36)=-38
-3	-24	-3+(-24)=-27
-4	-18	-4+(-18)=-22
-6	-12	-6+(-12)=-18
-8	-9	-8+(-9)=-17

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\n" ); document.write( "\n" ); document.write( "From this list we can see that 4 and 18 add up to 22 and multiply to 72\r
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"3x%5E2%2B22xy%2B24y%5E2\"$ , replace $\"22xy\"$ with $\"4xy%2B18xy\"$ (notice $\"4xy%2B18xy\"$ adds up to $\"22xy\"$ . So it is equivalent to $\"22xy\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"3x%5E2%2Bhighlight%284xy%2B18xy%29%2B24y%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"3x%5E2%2B4xy%2B18xy%2B24y%5E2\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%283x%5E2%2B4xy%29%2B%2818xy%2B24y%5E2%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"x%283x%2B4y%29%2B6y%283x%2B4y%29\"$ Factor out the GCF of $\"x\"$ out of the first group. Factor out the GCF of $\"6y\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%28x%2B6y%29%283x%2B4y%29\"$ Since we have a common term of $\"3x%2B4y\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"3x%5E2%2B4xy%2B18xy%2B24y%5E2\"$ factors to $\"%28x%2B6y%29%283x%2B4y%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So this also means that $\"3x%5E2%2B22xy%2B24y%5E2\"$ factors to $\"%28x%2B6y%29%283x%2B4y%29\"$ (since $\"3x%5E2%2B22xy%2B24y%5E2\"$ is equivalent to $\"3x%5E2%2B4xy%2B18xy%2B24y%5E2\"$ )\r
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\n" ); document.write( "\n" ); document.write( " Answer:\r
\n" ); document.write( "\n" ); document.write( "So $\"3x%5E2%2B22xy%2B24y%5E2\"$ factors to $\"%28x%2B6y%29%283x%2B4y%29\"$
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