document.write( "Question 183247: 5y^2-28y-12 \n" ); document.write( "

Algebra.Com's Answer #137585 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
I'm assuming that you want to factor this.\r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"5y%5E2-28y-12\"$ , we can see that the first coefficient is $\"5\"$ , the second coefficient is $\"-28\"$ , and the last term is $\"-12\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"5\"$ by the last term $\"-12\"$ to get $\"%285%29%28-12%29=-60\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-60\"$ (the previous product) and add to the second coefficient $\"-28\"$ ?\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 $\"-60\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-60\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,5,6,10,12,15,20,30,60\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-5,-6,-10,-12,-15,-20,-30,-60\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 $\"-60\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-60)
\n" ); document.write( "2*(-30)
\n" ); document.write( "3*(-20)
\n" ); document.write( "4*(-15)
\n" ); document.write( "5*(-12)
\n" ); document.write( "6*(-10)
\n" ); document.write( "(-1)*(60)
\n" ); document.write( "(-2)*(30)
\n" ); document.write( "(-3)*(20)
\n" ); document.write( "(-4)*(15)
\n" ); document.write( "(-5)*(12)
\n" ); document.write( "(-6)*(10)\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 $\"-28\"$ :\r
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First Number	Second Number	Sum
1	-60	1+(-60)=-59
2	-30	2+(-30)=-28
3	-20	3+(-20)=-17
4	-15	4+(-15)=-11
5	-12	5+(-12)=-7
6	-10	6+(-10)=-4
-1	60	-1+60=59
-2	30	-2+30=28
-3	20	-3+20=17
-4	15	-4+15=11
-5	12	-5+12=7
-6	10	-6+10=4

\r
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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"2\"$ and $\"-30\"$ add to $\"-28\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"2\"$ and $\"-30\"$ both multiply to $\"-60\"$ and add to $\"-28\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-28y\"$ with $\"2y-30y\"$ . Remember, $\"2\"$ and $\"-30\"$ add to $\"-28\"$ . So this shows us that $\"2y-30y=-28y\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"5y%5E2%2Bhighlight%282y-30y%29-12\"$ Replace the second term $\"-28y\"$ with $\"2y-30y\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%285y%5E2%2B2y%29%2B%28-30y-12%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"y%285y%2B2%29%2B%28-30y-12%29\"$ Factor out the GCF $\"y\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"y%285y%2B2%29-6%285y%2B2%29\"$ Factor out $\"6\"$ 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( " $\"%28y-6%29%285y%2B2%29\"$ Combine like terms. Or factor out the common term $\"5y%2B2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"5y%5E2-28y-12\"$ factors to $\"%28y-6%29%285y%2B2%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"5y%5E2-28y-12=%28y-6%29%285y%2B2%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by FOILing $\"%28y-6%29%285y%2B2%29\"$ to get $\"5y%5E2-28y-12\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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