document.write( "Question 385262: 12y^2+5y-3 \n" ); document.write( "

Algebra.Com's Answer #272577 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
I'm assuming you want to factor this.\r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"12y%5E2%2B5y-3\"$ , we can see that the first coefficient is $\"12\"$ , the second coefficient is $\"5\"$ , and the last term is $\"-3\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"12\"$ by the last term $\"-3\"$ to get $\"%2812%29%28-3%29=-36\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-36\"$ (the previous product) and add to the second coefficient $\"5\"$ ?\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 $\"-36\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-36\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,9,12,18,36\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-9,-12,-18,-36\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 $\"-36\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-36) = -36
\n" ); document.write( "2*(-18) = -36
\n" ); document.write( "3*(-12) = -36
\n" ); document.write( "4*(-9) = -36
\n" ); document.write( "6*(-6) = -36
\n" ); document.write( "(-1)*(36) = -36
\n" ); document.write( "(-2)*(18) = -36
\n" ); document.write( "(-3)*(12) = -36
\n" ); document.write( "(-4)*(9) = -36
\n" ); document.write( "(-6)*(6) = -36\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 $\"5\"$ :\r
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First Number	Second Number	Sum
1	-36	1+(-36)=-35
2	-18	2+(-18)=-16
3	-12	3+(-12)=-9
4	-9	4+(-9)=-5
6	-6	6+(-6)=0
-1	36	-1+36=35
-2	18	-2+18=16
-3	12	-3+12=9
-4	9	-4+9=5
-6	6	-6+6=0

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-4\"$ and $\"9\"$ add to $\"5\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-4\"$ and $\"9\"$ both multiply to $\"-36\"$ and add to $\"5\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"5y\"$ with $\"-4y%2B9y\"$ . Remember, $\"-4\"$ and $\"9\"$ add to $\"5\"$ . So this shows us that $\"-4y%2B9y=5y\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"12y%5E2%2Bhighlight%28-4y%2B9y%29-3\"$ Replace the second term $\"5y\"$ with $\"-4y%2B9y\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%2812y%5E2-4y%29%2B%289y-3%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"4y%283y-1%29%2B%289y-3%29\"$ Factor out the GCF $\"4y\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"4y%283y-1%29%2B3%283y-1%29\"$ 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( " $\"%284y%2B3%29%283y-1%29\"$ Combine like terms. Or factor out the common term $\"3y-1\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"12y%5E2%2B5y-3\"$ factors to $\"%284y%2B3%29%283y-1%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"12y%5E2%2B5y-3=%284y%2B3%29%283y-1%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%284y%2B3%29%283y-1%29\"$ to get $\"12y%5E2%2B5y-3\"$ 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, feel free to check out my tutoring website\r
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\n" ); document.write( "\n" ); document.write( "Jim \n" ); document.write( "

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