document.write( "Question 672867: 9y^2+17y-2 \n" ); document.write( "

Algebra.Com's Answer #418353 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"9y%5E2%2B17y-2\"$ , we can see that the first coefficient is $\"9\"$ , the second coefficient is $\"17\"$ , and the last term is $\"-2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"9\"$ by the last term $\"-2\"$ to get $\"%289%29%28-2%29=-18\"$ .\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 $\"17\"$ ?\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 $\"17\"$ :\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 $\"-1\"$ and $\"18\"$ add to $\"17\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-1\"$ and $\"18\"$ both multiply to $\"-18\"$ and add to $\"17\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"17y\"$ with $\"-y%2B18y\"$ . Remember, $\"-1\"$ and $\"18\"$ add to $\"17\"$ . So this shows us that $\"-y%2B18y=17y\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"9y%5E2%2Bhighlight%28-y%2B18y%29-2\"$ Replace the second term $\"17y\"$ with $\"-y%2B18y\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%289y%5E2-y%29%2B%2818y-2%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"y%289y-1%29%2B%2818y-2%29\"$ Factor out the GCF $\"y\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"y%289y-1%29%2B2%289y-1%29\"$ Factor out $\"2\"$ 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( " $\"%28y%2B2%29%289y-1%29\"$ Combine like terms. Or factor out the common term $\"9y-1\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"9y%5E2%2B17y-2\"$ factors to $\"%28y%2B2%29%289y-1%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"9y%5E2%2B17y-2=%28y%2B2%29%289y-1%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%28y%2B2%29%289y-1%29\"$ to get $\"9y%5E2%2B17y-2\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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