document.write( "Question 167669: factor completely
\n" ); document.write( "6a^2-29a+9 \n" ); document.write( "

Algebra.Com's Answer #123570 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"6a%5E2-29a%2B9\"$ , we can see that the first coefficient is $\"6\"$ , the second coefficient is $\"-29\"$ , and the last term is $\"9\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"6\"$ by the last term $\"9\"$ to get $\"%286%29%289%29=54\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"54\"$ (the previous product) and add to the second coefficient $\"-29\"$ ?\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 $\"54\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"54\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,6,9,18,27,54\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-6,-9,-18,-27,-54\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 $\"54\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*54
\n" ); document.write( "2*27
\n" ); document.write( "3*18
\n" ); document.write( "6*9
\n" ); document.write( "(-1)*(-54)
\n" ); document.write( "(-2)*(-27)
\n" ); document.write( "(-3)*(-18)
\n" ); document.write( "(-6)*(-9)\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 $\"-29\"$ :\r
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First Number	Second Number	Sum
1	54	1+54=55
2	27	2+27=29
3	18	3+18=21
6	9	6+9=15
-1	-54	-1+(-54)=-55
-2	-27	-2+(-27)=-29
-3	-18	-3+(-18)=-21
-6	-9	-6+(-9)=-15

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-2\"$ and $\"-27\"$ add to $\"-29\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-2\"$ and $\"-27\"$ both multiply to $\"54\"$ and add to $\"-29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-29a\"$ with $\"-2a-27a\"$ . Remember, $\"-2\"$ and $\"-27\"$ add to $\"-29\"$ . So this shows us that $\"-2a-27a=-29a\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"6a%5E2%2Bhighlight%28-2a-27a%29%2B9\"$ Replace the second term $\"-29a\"$ with $\"-2a-27a\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%286a%5E2-2a%29%2B%28-27a%2B9%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"2a%283a-1%29%2B%28-27a%2B9%29\"$ Factor out the GCF $\"2a\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"2a%283a-1%29-9%283a-1%29\"$ Factor out $\"9\"$ 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( " $\"%282a-9%29%283a-1%29\"$ Combine like terms. Or factor out the common term $\"3a-1\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"6a%5E2-29a%2B9\"$ factors to $\"%282a-9%29%283a-1%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by FOILing $\"%282a-9%29%283a-1%29\"$ to get $\"6a%5E2-29a%2B9\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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