document.write( "Question 117381: Factor Completely.\r
\n" ); document.write( "\n" ); document.write( "9x^2+12xy+4y^2-25.\r
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Algebra.Com's Answer #85389 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
First let's factor $\"9x%5E2%2B12xy%2B4y%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at $\"9x%5E2%2B12xy%2B4y%5E2\"$ we can see that the first term is $\"9x%5E2\"$ and the last term is $\"4y%5E2\"$ where the coefficients are 9 and 4 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 9 and the last coefficient 4 to get 36. Now what two numbers multiply to 36 and add to the middle coefficient 12? Let's list all of the factors of 36:\r
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\n" ); document.write( "\n" ); document.write( "Factors of 36:\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-9,-12,-18 ...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 36\r
\n" ); document.write( "\n" ); document.write( "1*36\r
\n" ); document.write( "\n" ); document.write( "2*18\r
\n" ); document.write( "\n" ); document.write( "3*12\r
\n" ); document.write( "\n" ); document.write( "4*9\r
\n" ); document.write( "\n" ); document.write( "6*6\r
\n" ); document.write( "\n" ); document.write( "(-1)*(-36)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(-18)\r
\n" ); document.write( "\n" ); document.write( "(-3)*(-12)\r
\n" ); document.write( "\n" ); document.write( "(-4)*(-9)\r
\n" ); document.write( "\n" ); document.write( "(-6)*(-6)\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 12? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 12\r
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First Number	Second Number	Sum
1	36	1+36=37
2	18	2+18=20
3	12	3+12=15
4	9	4+9=13
6	6	6+6=12
-1	-36	-1+(-36)=-37
-2	-18	-2+(-18)=-20
-3	-12	-3+(-12)=-15
-4	-9	-4+(-9)=-13
-6	-6	-6+(-6)=-12

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\n" ); document.write( "\n" ); document.write( "From this list we can see that 6 and 6 add up to 12 and multiply to 36\r
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"9x%5E2%2B12xy%2B4y%5E2\"$ , replace $\"12xy\"$ with $\"6xy%2B6xy\"$ (notice $\"6xy%2B6xy\"$ adds up to $\"12xy\"$ . So it is equivalent to $\"12xy\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"9x%5E2%2Bhighlight%286xy%2B6xy%29%2B4y%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"9x%5E2%2B6xy%2B6xy%2B4y%5E2\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%289x%5E2%2B6xy%29%2B%286xy%2B4y%5E2%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"3x%283x%2B2y%29%2B2y%283x%2B2y%29\"$ Factor out the GCF of $\"3x\"$ out of the first group. Factor out the GCF of $\"2y\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%283x%2B2y%29%283x%2B2y%29\"$ Since we have a common term of $\"3x%2B2y\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"9x%5E2%2B12xy%2B4y%5E2\"$ factors to $\"%283x%2B2y%29%283x%2B2y%29\"$ which also looks like $\"%283x%2B2y%29%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "So we now have\r
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\n" ); document.write( "\n" ); document.write( " $\"%283x%2B2y%29%5E2-25\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now factor the expression using the difference of squares\r
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\n" ); document.write( "\n" ); document.write( " $\"%283x%2B2y%2B5%29%283x%2B2y-5%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"9x%5E2%2B12xy%2B4y%5E2-25\"$ factors to $\"%283x%2B2y%2B5%29%283x%2B2y-5%29\"$ \n" ); document.write( "

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