document.write( "Question 139519: What do I solve this by factoring?\r
\n" ); document.write( "\n" ); document.write( "20x^2 + 13x + 2\r
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Algebra.Com's Answer #101720 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( "Looking at $\"20x%5E2%2B13x%2B2\"$ we can see that the first term is $\"20x%5E2\"$ and the last term is $\"2\"$ where the coefficients are 20 and 2 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 20 and the last coefficient 2 to get 40. Now what two numbers multiply to 40 and add to the middle coefficient 13? Let's list all of the factors of 40:\r
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\n" ); document.write( "\n" ); document.write( "Factors of 40:\r
\n" ); document.write( "\n" ); document.write( "1,2,4,5,8,10,20,40\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-4,-5,-8,-10,-20,-40 ...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 40\r
\n" ); document.write( "\n" ); document.write( "1*40\r
\n" ); document.write( "\n" ); document.write( "2*20\r
\n" ); document.write( "\n" ); document.write( "4*10\r
\n" ); document.write( "\n" ); document.write( "5*8\r
\n" ); document.write( "\n" ); document.write( "(-1)*(-40)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(-20)\r
\n" ); document.write( "\n" ); document.write( "(-4)*(-10)\r
\n" ); document.write( "\n" ); document.write( "(-5)*(-8)\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 13? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 13\r
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First Number	Second Number	Sum
1	40	1+40=41
2	20	2+20=22
4	10	4+10=14
5	8	5+8=13
-1	-40	-1+(-40)=-41
-2	-20	-2+(-20)=-22
-4	-10	-4+(-10)=-14
-5	-8	-5+(-8)=-13

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\n" ); document.write( "\n" ); document.write( "From this list we can see that 5 and 8 add up to 13 and multiply to 40\r
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"20x%5E2%2B13x%2B2\"$ , replace $\"13x\"$ with $\"5x%2B8x\"$ (notice $\"5x%2B8x\"$ adds up to $\"13x\"$ . So it is equivalent to $\"13x\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"20x%5E2%2Bhighlight%285x%2B8x%29%2B2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"20x%5E2%2B5x%2B8x%2B2\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%2820x%5E2%2B5x%29%2B%288x%2B2%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"5x%284x%2B1%29%2B2%284x%2B1%29\"$ Factor out the GCF of $\"5x\"$ out of the first group. Factor out the GCF of $\"2\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%285x%2B2%29%284x%2B1%29\"$ Since we have a common term of $\"4x%2B1\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"20x%5E2%2B5x%2B8x%2B2\"$ factors to $\"%285x%2B2%29%284x%2B1%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So this also means that $\"20x%5E2%2B13x%2B2\"$ factors to $\"%285x%2B2%29%284x%2B1%29\"$ (since $\"20x%5E2%2B13x%2B2\"$ is equivalent to $\"20x%5E2%2B5x%2B8x%2B2\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"%285x%2B2%29%284x%2B1%29=0\"$ Set the factorization equal to zero\r
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\n" ); document.write( "\n" ); document.write( "Now set each factor equal to zero:\r
\n" ); document.write( "\n" ); document.write( " $\"5x%2B2=0\"$ or $\"4x%2B1=0\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"x=-2%2F5\"$ or $\"x=-1%2F4\"$ Now solve for x in each case\r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So our solutions are\r
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\n" ); document.write( "\n" ); document.write( " $\"x=-2%2F5\"$ or $\"x=-1%2F4\"$ \r
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