document.write( "Question 173291: Can you please explain the following in full depth:\r
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Algebra.Com's Answer #128119 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 $\"3x%5E2%2B26xy%2B16y%5E2\"$ we can see that the first term is $\"3x%5E2\"$ and the last term is $\"16y%5E2\"$ where the coefficients are 3 and 16 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 3 and the last coefficient 16 to get 48. Now what two numbers multiply to 48 and add to the middle coefficient 26? Let's list all of the factors of 48:\r
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\n" ); document.write( "\n" ); document.write( "Factors of 48:\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-8,-12,-16,-24,-48 ...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 48\r
\n" ); document.write( "\n" ); document.write( "1*48\r
\n" ); document.write( "\n" ); document.write( "2*24\r
\n" ); document.write( "\n" ); document.write( "3*16\r
\n" ); document.write( "\n" ); document.write( "4*12\r
\n" ); document.write( "\n" ); document.write( "6*8\r
\n" ); document.write( "\n" ); document.write( "(-1)*(-48)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(-24)\r
\n" ); document.write( "\n" ); document.write( "(-3)*(-16)\r
\n" ); document.write( "\n" ); document.write( "(-4)*(-12)\r
\n" ); document.write( "\n" ); document.write( "(-6)*(-8)\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 26? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 26\r
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First Number	Second Number	Sum
1	48	1+48=49
2	24	2+24=26
3	16	3+16=19
4	12	4+12=16
6	8	6+8=14
-1	-48	-1+(-48)=-49
-2	-24	-2+(-24)=-26
-3	-16	-3+(-16)=-19
-4	-12	-4+(-12)=-16
-6	-8	-6+(-8)=-14

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\n" ); document.write( "\n" ); document.write( "From this list we can see that 2 and 24 add up to 26 and multiply to 48\r
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"3x%5E2%2B26xy%2B16y%5E2\"$ , replace $\"26xy\"$ with $\"2xy%2B24xy\"$ (notice $\"2xy%2B24xy\"$ adds up to $\"26xy\"$ . So it is equivalent to $\"26xy\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"3x%5E2%2Bhighlight%282xy%2B24xy%29%2B16y%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"3x%5E2%2B2xy%2B24xy%2B16y%5E2\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%283x%5E2%2B2xy%29%2B%2824xy%2B16y%5E2%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"x%283x%2B2y%29%2B8y%283x%2B2y%29\"$ Factor out the GCF of $\"x\"$ out of the first group. Factor out the GCF of $\"8y\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%28x%2B8y%29%283x%2B2y%29\"$ 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 $\"3x%5E2%2B2xy%2B24xy%2B16y%5E2\"$ factors to $\"%28x%2B8y%29%283x%2B2y%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So this also means that $\"3x%5E2%2B26xy%2B16y%5E2\"$ factors to $\"%28x%2B8y%29%283x%2B2y%29\"$ (since $\"3x%5E2%2B26xy%2B16y%5E2\"$ is equivalent to $\"3x%5E2%2B2xy%2B24xy%2B16y%5E2\"$ )\r
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\n" ); document.write( "\n" ); document.write( " Answer:\r
\n" ); document.write( "\n" ); document.write( "So $\"3x%5E2%2B26xy%2B16y%5E2\"$ factors to $\"%28x%2B8y%29%283x%2B2y%29\"$
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