document.write( "Question 332267: Please help me with this equation, my book does not cover this fully and I am having a hard time understanding. I need to factor the perfect square expression. The problem is x^2 + 12xy + 36y^2 \n" ); document.write( "

Algebra.Com's Answer #238125 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "Looking at the expression $\"x%5E2%2B12xy%2B36y%5E2\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"12\"$ , and the last coefficient is $\"36\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last coefficient $\"36\"$ to get $\"%281%29%2836%29=36\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"36\"$ (the previous product) and add to the second coefficient $\"12\"$ ?\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 $\"36\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"36\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,9,12,18,36\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-9,-12,-18,-36\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 $\"36\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*36 = 36
\n" ); document.write( "2*18 = 36
\n" ); document.write( "3*12 = 36
\n" ); document.write( "4*9 = 36
\n" ); document.write( "6*6 = 36
\n" ); document.write( "(-1)*(-36) = 36
\n" ); document.write( "(-2)*(-18) = 36
\n" ); document.write( "(-3)*(-12) = 36
\n" ); document.write( "(-4)*(-9) = 36
\n" ); document.write( "(-6)*(-6) = 36\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 $\"12\"$ :\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 the table, we can see that the two numbers $\"6\"$ and $\"6\"$ add to $\"12\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"6\"$ and $\"6\"$ both multiply to $\"36\"$ and add to $\"12\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"12xy\"$ with $\"6xy%2B6xy\"$ . Remember, $\"6\"$ and $\"6\"$ add to $\"12\"$ . So this shows us that $\"6xy%2B6xy=12xy\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"x%5E2%2Bhighlight%286xy%2B6xy%29%2B36y%5E2\"$ Replace the second term $\"12xy\"$ with $\"6xy%2B6xy\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%28x%5E2%2B6xy%29%2B%286xy%2B36y%5E2%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x%2B6y%29%2B%286xy%2B36y%5E2%29\"$ Factor out the GCF $\"x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x%2B6y%29%2B6y%28x%2B6y%29\"$ Factor out $\"6y\"$ 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( " $\"%28x%2B6y%29%28x%2B6y%29\"$ Combine like terms. Or factor out the common term $\"x%2B6y\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"%28x%2B6y%29%5E2\"$ Condense the terms.\r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"x%5E2%2B12xy%2B36y%5E2\"$ factors to $\"%28x%2B6y%29%5E2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"x%5E2%2B12xy%2B36y%5E2=%28x%2B6y%29%5E2\"$ .\r
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