document.write( "Question 609785: can you help me solve the equation by factoring
\n" ); document.write( "x^2-21x+108=0 \n" ); document.write( "

Algebra.Com's Answer #383943 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
I'll help you factor this. Once the factorization is done, I'll let you take over.\r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"x%5E2-21x%2B108\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"-21\"$ , and the last term is $\"108\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last term $\"108\"$ to get $\"%281%29%28108%29=108\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"108\"$ (the previous product) and add to the second coefficient $\"-21\"$ ?\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 $\"108\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"108\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,9,12,18,27,36,54,108\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-9,-12,-18,-27,-36,-54,-108\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 $\"108\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*108 = 108
\n" ); document.write( "2*54 = 108
\n" ); document.write( "3*36 = 108
\n" ); document.write( "4*27 = 108
\n" ); document.write( "6*18 = 108
\n" ); document.write( "9*12 = 108
\n" ); document.write( "(-1)*(-108) = 108
\n" ); document.write( "(-2)*(-54) = 108
\n" ); document.write( "(-3)*(-36) = 108
\n" ); document.write( "(-4)*(-27) = 108
\n" ); document.write( "(-6)*(-18) = 108
\n" ); document.write( "(-9)*(-12) = 108\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 $\"-21\"$ :\r
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First Number	Second Number	Sum
1	108	1+108=109
2	54	2+54=56
3	36	3+36=39
4	27	4+27=31
6	18	6+18=24
9	12	9+12=21
-1	-108	-1+(-108)=-109
-2	-54	-2+(-54)=-56
-3	-36	-3+(-36)=-39
-4	-27	-4+(-27)=-31
-6	-18	-6+(-18)=-24
-9	-12	-9+(-12)=-21

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-9\"$ and $\"-12\"$ add to $\"-21\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-9\"$ and $\"-12\"$ both multiply to $\"108\"$ and add to $\"-21\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-21x\"$ with $\"-9x-12x\"$ . Remember, $\"-9\"$ and $\"-12\"$ add to $\"-21\"$ . So this shows us that $\"-9x-12x=-21x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"x%5E2%2Bhighlight%28-9x-12x%29%2B108\"$ Replace the second term $\"-21x\"$ with $\"-9x-12x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%28x%5E2-9x%29%2B%28-12x%2B108%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x-9%29%2B%28-12x%2B108%29\"$ Factor out the GCF $\"x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x-9%29-12%28x-9%29\"$ Factor out $\"12\"$ 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( " $\"%28x-12%29%28x-9%29\"$ Combine like terms. Or factor out the common term $\"x-9\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"x%5E2-21x%2B108\"$ factors to $\"%28x-12%29%28x-9%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"x%5E2-21x%2B108=%28x-12%29%28x-9%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%28x-12%29%28x-9%29\"$ to get $\"x%5E2-21x%2B108\"$ or by graphing the original expression and the answer (the two graphs should be identical).\r
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\n" ); document.write( "\n" ); document.write( "So this means $\"x%5E2-21x%2B108=0\"$ turns into $\"%28x-12%29%28x-9%29=0\"$ \r
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\n" ); document.write( "If you need more help, email me at jim_thompson5910@hotmail.com\r
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\n" ); document.write( "\n" ); document.write( "Also, please consider visiting my website: http://www.freewebs.com/jimthompson5910/home.html and making a donation. Thank you\r
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