document.write( "Question 133704: how do you factor this problem: 6n^2 - 19n+ 10
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Algebra.Com's Answer #97809 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at $\"6n%5E2-19n%2B10\"$ we can see that the first term is $\"6n%5E2\"$ and the last term is $\"10\"$ where the coefficients are 6 and 10 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 6 and the last coefficient 10 to get 60. Now what two numbers multiply to 60 and add to the middle coefficient -19? Let's list all of the factors of 60:\r
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\n" ); document.write( "\n" ); document.write( "Factors of 60:\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,5,6,10,12,15,20,30,60\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-5,-6,-10,-12,-15,-20,-30,-60 ...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 60\r
\n" ); document.write( "\n" ); document.write( "1*60\r
\n" ); document.write( "\n" ); document.write( "2*30\r
\n" ); document.write( "\n" ); document.write( "3*20\r
\n" ); document.write( "\n" ); document.write( "4*15\r
\n" ); document.write( "\n" ); document.write( "5*12\r
\n" ); document.write( "\n" ); document.write( "6*10\r
\n" ); document.write( "\n" ); document.write( "(-1)*(-60)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(-30)\r
\n" ); document.write( "\n" ); document.write( "(-3)*(-20)\r
\n" ); document.write( "\n" ); document.write( "(-4)*(-15)\r
\n" ); document.write( "\n" ); document.write( "(-5)*(-12)\r
\n" ); document.write( "\n" ); document.write( "(-6)*(-10)\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 -19? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -19\r
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First Number	Second Number	Sum
1	60	1+60=61
2	30	2+30=32
3	20	3+20=23
4	15	4+15=19
5	12	5+12=17
6	10	6+10=16
-1	-60	-1+(-60)=-61
-2	-30	-2+(-30)=-32
-3	-20	-3+(-20)=-23
-4	-15	-4+(-15)=-19
-5	-12	-5+(-12)=-17
-6	-10	-6+(-10)=-16

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\n" ); document.write( "\n" ); document.write( "From this list we can see that -4 and -15 add up to -19 and multiply to 60\r
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"6n%5E2-19n%2B10\"$ , replace $\"-19n\"$ with $\"-4n%2B-15n\"$ (notice $\"-4n%2B-15n\"$ adds up to $\"-19n\"$ . So it is equivalent to $\"-19n\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"6n%5E2%2Bhighlight%28-4n%2B-15n%29%2B10\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"6n%5E2-4n-15n%2B10\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%286n%5E2-4n%29%2B%28-15n%2B10%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"2n%283n-2%29-5%283n-2%29\"$ Factor out the GCF of $\"2n\"$ out of the first group. Factor out the GCF of $\"-5\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%282n-5%29%283n-2%29\"$ Since we have a common term of $\"3n-2\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"6n%5E2-4n-15n%2B10\"$ factors to $\"%282n-5%29%283n-2%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So this also means that $\"6n%5E2-19n%2B10\"$ factors to $\"%282n-5%29%283n-2%29\"$ (since $\"6n%5E2-19n%2B10\"$ is equivalent to $\"6n%5E2-4n-15n%2B10\"$ )\r
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\n" ); document.write( "\n" ); document.write( " Answer:\r
\n" ); document.write( "\n" ); document.write( "So $\"6n%5E2-19n%2B10\"$ factors to $\"%282n-5%29%283n-2%29\"$
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