document.write( "Question 905066: hi can you help me figure this out ;Factor using the ac-method. \n" ); document.write( "

Algebra.Com's Answer #549022 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( " $\"108x%5E2%2B63xy%2B9y%5E2\"$ Start with the given expression.\r
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\n" ); document.write( "\n" ); document.write( " $\"9%2812x%5E2%2B7xy%2By%5E2%29\"$ Factor out the GCF $\"9\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now let's try to factor the inner expression $\"12x%5E2%2B7xy%2By%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"12x%5E2%2B7xy%2By%5E2\"$ , we can see that the first coefficient is $\"12\"$ , the second coefficient is $\"7\"$ , and the last coefficient is $\"1\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"12\"$ by the last coefficient $\"1\"$ to get $\"%2812%29%281%29=12\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"12\"$ (the previous product) and add to the second coefficient $\"7\"$ ?\r
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\n" ); document.write( "\n" ); document.write( "To find these two numbers, we need to list all of the factors of $\"12\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"12\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,12\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-12\r
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\n" ); document.write( "\n" ); document.write( "Note: list the negative of each factor. 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 $\"12\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*12 = 12
\n" ); document.write( "2*6 = 12
\n" ); document.write( "3*4 = 12
\n" ); document.write( "(-1)*(-12) = 12
\n" ); document.write( "(-2)*(-6) = 12
\n" ); document.write( "(-3)*(-4) = 12\r
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\n" ); document.write( "\n" ); document.write( "Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"7\"$ :\r
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First Number	Second Number	Sum
1	12	1+12=13
2	6	2+6=8
3	4	3+4=7
-1	-12	-1+(-12)=-13
-2	-6	-2+(-6)=-8
-3	-4	-3+(-4)=-7

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"3\"$ and $\"4\"$ add to $\"7\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"3\"$ and $\"4\"$ both multiply to $\"12\"$ and add to $\"7\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"7xy\"$ with $\"3xy%2B4xy\"$ . Remember, $\"3\"$ and $\"4\"$ add to $\"7\"$ . So this shows us that $\"3xy%2B4xy=7xy\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"12x%5E2%2Bhighlight%283xy%2B4xy%29%2By%5E2\"$ Replace the second term $\"7xy\"$ with $\"3xy%2B4xy\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%2812x%5E2%2B3xy%29%2B%284xy%2By%5E2%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"3x%284x%2By%29%2B%284xy%2By%5E2%29\"$ Factor out the GCF $\"3x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"3x%284x%2By%29%2By%284x%2By%29\"$ Factor out $\"y\"$ from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.\r
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\n" ); document.write( "\n" ); document.write( " $\"%283x%2By%29%284x%2By%29\"$ Combine like terms. Or factor out the common term $\"4x%2By\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"9%2812x%5E2%2B7xy%2By%5E2%29\"$ then factors further to $\"9%283x%2By%29%284x%2By%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"108x%5E2%2B63xy%2B9y%5E2\"$ completely factors to $\"9%283x%2By%29%284x%2By%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"108x%5E2%2B63xy%2B9y%5E2=9%283x%2By%29%284x%2By%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"9%283x%2By%29%284x%2By%29\"$ to get $\"108x%5E2%2B63xy%2B9y%5E2\"$ .
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