document.write( "Question 783452: 6x^2-7x-3 \n" ); document.write( "

Algebra.Com's Answer #476788 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"6x%5E2-7x-3\"$ , we can see that the first coefficient is $\"6\"$ , the second coefficient is $\"-7\"$ , and the last term is $\"-3\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"6\"$ by the last term $\"-3\"$ to get $\"%286%29%28-3%29=-18\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-18\"$ (the previous product) and add to the second coefficient $\"-7\"$ ?\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 $\"-18\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-18\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,6,9,18\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-6,-9,-18\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 $\"-18\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-18) = -18
\n" ); document.write( "2*(-9) = -18
\n" ); document.write( "3*(-6) = -18
\n" ); document.write( "(-1)*(18) = -18
\n" ); document.write( "(-2)*(9) = -18
\n" ); document.write( "(-3)*(6) = -18\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 $\"-7\"$ :\r
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First Number	Second Number	Sum
1	-18	1+(-18)=-17
2	-9	2+(-9)=-7
3	-6	3+(-6)=-3
-1	18	-1+18=17
-2	9	-2+9=7
-3	6	-3+6=3

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"2\"$ and $\"-9\"$ add to $\"-7\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"2\"$ and $\"-9\"$ both multiply to $\"-18\"$ and add to $\"-7\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-7x\"$ with $\"2x-9x\"$ . Remember, $\"2\"$ and $\"-9\"$ add to $\"-7\"$ . So this shows us that $\"2x-9x=-7x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"6x%5E2%2Bhighlight%282x-9x%29-3\"$ Replace the second term $\"-7x\"$ with $\"2x-9x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%286x%5E2%2B2x%29%2B%28-9x-3%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"2x%283x%2B1%29%2B%28-9x-3%29\"$ Factor out the GCF $\"2x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"2x%283x%2B1%29-3%283x%2B1%29\"$ Factor out $\"3\"$ 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( " $\"%282x-3%29%283x%2B1%29\"$ Combine like terms. Or factor out the common term $\"3x%2B1\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"6x%5E2-7x-3\"$ factors to $\"%282x-3%29%283x%2B1%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"6x%5E2-7x-3=%282x-3%29%283x%2B1%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%282x-3%29%283x%2B1%29\"$ to get $\"6x%5E2-7x-3\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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