document.write( "Question 182121: Can you help me solve this? Factor completely. 12x^2 - 10x -8
\n" ); document.write( "I keep getting confused because I'm supposed to find terms that have a product of negative 8 and a sum of negative 10, but all of the answers doesn't factor back out for me. \n" ); document.write( "

Algebra.Com's Answer #136691 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( " $\"12x%5E2-10x-8\"$ Start with the given expression\r
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\n" ); document.write( "\n" ); document.write( " $\"2%286x%5E2-5x-4%29\"$ Factor out the GCF $\"2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's focus on the inner expression $\"6x%5E2-5x-4\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at $\"6x%5E2-5x-4\"$ we can see that the first term is $\"6x%5E2\"$ and the last term is $\"-4\"$ where the coefficients are 6 and -4 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 6 and the last coefficient -4 to get -24. Now what two numbers multiply to -24 and add to the middle coefficient -5? Let's list all of the factors of -24:\r
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\n" ); document.write( "\n" ); document.write( "Factors of -24:\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,8,12,24\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-8,-12,-24 ...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 -24\r
\n" ); document.write( "\n" ); document.write( "(1)*(-24)\r
\n" ); document.write( "\n" ); document.write( "(2)*(-12)\r
\n" ); document.write( "\n" ); document.write( "(3)*(-8)\r
\n" ); document.write( "\n" ); document.write( "(4)*(-6)\r
\n" ); document.write( "\n" ); document.write( "(-1)*(24)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(12)\r
\n" ); document.write( "\n" ); document.write( "(-3)*(8)\r
\n" ); document.write( "\n" ); document.write( "(-4)*(6)\r
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\n" ); document.write( "\n" ); document.write( "note: remember, the product of a negative and a positive number is a negative number\r
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\n" ); document.write( "\n" ); document.write( "Now which of these pairs add to -5? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -5\r
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First Number	Second Number	Sum
1	-24	1+(-24)=-23
2	-12	2+(-12)=-10
3	-8	3+(-8)=-5
4	-6	4+(-6)=-2
-1	24	-1+24=23
-2	12	-2+12=10
-3	8	-3+8=5
-4	6	-4+6=2

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\n" ); document.write( "\n" ); document.write( "From this list we can see that 3 and -8 add up to -5 and multiply to -24\r
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"6x%5E2-5x-4\"$ , replace $\"-5x\"$ with $\"3x%2B-8x\"$ (notice $\"3x%2B-8x\"$ adds up to $\"-5x\"$ . So it is equivalent to $\"-5x\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"6x%5E2%2Bhighlight%283x%2B-8x%29%2B-4\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"6x%5E2%2B3x-8x-4\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%286x%5E2%2B3x%29%2B%28-8x-4%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"3x%282x%2B1%29-4%282x%2B1%29\"$ Factor out the GCF of $\"3x\"$ out of the first group. Factor out the GCF of $\"-4\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%283x-4%29%282x%2B1%29\"$ Since we have a common term of $\"2x%2B1\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"6x%5E2%2B3x-8x-4\"$ factors to $\"%283x-4%29%282x%2B1%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So this also means that $\"6x%5E2-5x-4\"$ factors to $\"%283x-4%29%282x%2B1%29\"$ (since $\"6x%5E2-5x-4\"$ is equivalent to $\"6x%5E2%2B3x-8x-4\"$ )\r
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\n" ); document.write( "\n" ); document.write( "So our expression goes from $\"2%286x%5E2-5x-4%29\"$ and factors further to $\"2%283x-4%29%282x%2B1%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"12x%5E2-10x-8\"$ factors to $\"2%283x-4%29%282x%2B1%29\"$ \r
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