document.write( "Question 119708: How do I factor this?\r
\n" ); document.write( "\n" ); document.write( "X^2-12x+36 \n" ); document.write( "

Algebra.Com's Answer #87709 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at $\"x%5E2-12x%2B36\"$ we can see that the first term is $\"x%5E2\"$ and the last term is $\"36\"$ where the coefficients are 1 and 36 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 1 and the last coefficient 36 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
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,9,12,18\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 $\"x%5E2-12x%2B36\"$ , replace $\"-12x\"$ with $\"-6x%2B-6x\"$ (notice $\"-6x%2B-6x\"$ adds up to $\"-12x\"$ . So it is equivalent to $\"-12x\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"x%5E2%2Bhighlight%28-6x%2B-6x%29%2B36\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"x%5E2-6x-6x%2B36\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%28x%5E2-6x%29%2B%28-6x%2B36%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x-6%29-6%28x-6%29\"$ Factor out the GCF of $\"x\"$ out of the first group. Factor out the GCF of $\"-6\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%28x-6%29%28x-6%29\"$ Since we have a common term of $\"x-6\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"x%5E2-6x-6x%2B36\"$ factors to $\"%28x-6%29%28x-6%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So this also means that $\"x%5E2-12x%2B36\"$ factors to $\"%28x-6%29%28x-6%29\"$ (since $\"x%5E2-12x%2B36\"$ is equivalent to $\"x%5E2-6x-6x%2B36\"$ )\r
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\n" ); document.write( "\n" ); document.write( "note: $\"%28x-6%29%28x-6%29\"$ is equivalent to $\"%28x-6%29%5E2\"$ since the term $\"x-6\"$ occurs twice. So $\"x%5E2-12x%2B36\"$ also factors to $\"%28x-6%29%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( " Answer:\r
\n" ); document.write( "\n" ); document.write( "So $\"x%5E2-12x%2B36\"$ factors to $\"%28x-6%29%5E2\"$
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