document.write( "Question 565246: 18p^2+9p-14 \n" ); document.write( "

Algebra.Com's Answer #365710 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"18p%5E2%2B9p-14\"$ , we can see that the first coefficient is $\"18\"$ , the second coefficient is $\"9\"$ , and the last term is $\"-14\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"18\"$ by the last term $\"-14\"$ to get $\"%2818%29%28-14%29=-252\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-252\"$ (the previous product) and add to the second coefficient $\"9\"$ ?\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 $\"-252\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-252\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,7,9,12,14,18,21,28,36,42,63,84,126,252\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-7,-9,-12,-14,-18,-21,-28,-36,-42,-63,-84,-126,-252\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 $\"-252\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-252) = -252
\n" ); document.write( "2*(-126) = -252
\n" ); document.write( "3*(-84) = -252
\n" ); document.write( "4*(-63) = -252
\n" ); document.write( "6*(-42) = -252
\n" ); document.write( "7*(-36) = -252
\n" ); document.write( "9*(-28) = -252
\n" ); document.write( "12*(-21) = -252
\n" ); document.write( "14*(-18) = -252
\n" ); document.write( "(-1)*(252) = -252
\n" ); document.write( "(-2)*(126) = -252
\n" ); document.write( "(-3)*(84) = -252
\n" ); document.write( "(-4)*(63) = -252
\n" ); document.write( "(-6)*(42) = -252
\n" ); document.write( "(-7)*(36) = -252
\n" ); document.write( "(-9)*(28) = -252
\n" ); document.write( "(-12)*(21) = -252
\n" ); document.write( "(-14)*(18) = -252\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 $\"9\"$ :\r
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First Number	Second Number	Sum
1	-252	1+(-252)=-251
2	-126	2+(-126)=-124
3	-84	3+(-84)=-81
4	-63	4+(-63)=-59
6	-42	6+(-42)=-36
7	-36	7+(-36)=-29
9	-28	9+(-28)=-19
12	-21	12+(-21)=-9
14	-18	14+(-18)=-4
-1	252	-1+252=251
-2	126	-2+126=124
-3	84	-3+84=81
-4	63	-4+63=59
-6	42	-6+42=36
-7	36	-7+36=29
-9	28	-9+28=19
-12	21	-12+21=9
-14	18	-14+18=4

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-12\"$ and $\"21\"$ add to $\"9\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-12\"$ and $\"21\"$ both multiply to $\"-252\"$ and add to $\"9\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"9p\"$ with $\"-12p%2B21p\"$ . Remember, $\"-12\"$ and $\"21\"$ add to $\"9\"$ . So this shows us that $\"-12p%2B21p=9p\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"18p%5E2%2Bhighlight%28-12p%2B21p%29-14\"$ Replace the second term $\"9p\"$ with $\"-12p%2B21p\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%2818p%5E2-12p%29%2B%2821p-14%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"6p%283p-2%29%2B%2821p-14%29\"$ Factor out the GCF $\"6p\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"6p%283p-2%29%2B7%283p-2%29\"$ Factor out $\"7\"$ 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( " $\"%286p%2B7%29%283p-2%29\"$ Combine like terms. Or factor out the common term $\"3p-2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"18p%5E2%2B9p-14\"$ factors to $\"%286p%2B7%29%283p-2%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"18p%5E2%2B9p-14=%286p%2B7%29%283p-2%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%286p%2B7%29%283p-2%29\"$ to get $\"18p%5E2%2B9p-14\"$ 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( "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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\n" ); document.write( "\n" ); document.write( "Jim\r
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