document.write( "Question 243323: I need help solving trinomials, sorry if I chose the wrong category
\n" ); document.write( "here is the problem: -y^2+10y-24. \n" ); document.write( "

Algebra.Com's Answer #178254 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
I'm assuming that you want to factor.\r
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\n" ); document.write( "\n" ); document.write( " $\"-y%5E2%2B10y-24\"$ Start with the given expression.\r
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\n" ); document.write( "\n" ); document.write( " $\"-%28y%5E2-10y%2B24%29\"$ Factor out the GCF $\"-1\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now let's try to factor the inner expression $\"y%5E2-10y%2B24\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"y%5E2-10y%2B24\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"-10\"$ , and the last term is $\"24\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last term $\"24\"$ to get $\"%281%29%2824%29=24\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"24\"$ (the previous product) and add to the second coefficient $\"-10\"$ ?\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 $\"24\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"24\"$ :\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-8,-12,-24\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 $\"24\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*24 = 24
\n" ); document.write( "2*12 = 24
\n" ); document.write( "3*8 = 24
\n" ); document.write( "4*6 = 24
\n" ); document.write( "(-1)*(-24) = 24
\n" ); document.write( "(-2)*(-12) = 24
\n" ); document.write( "(-3)*(-8) = 24
\n" ); document.write( "(-4)*(-6) = 24\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 $\"-10\"$ :\r
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First Number	Second Number	Sum
1	24	1+24=25
2	12	2+12=14
3	8	3+8=11
4	6	4+6=10
-1	-24	-1+(-24)=-25
-2	-12	-2+(-12)=-14
-3	-8	-3+(-8)=-11
-4	-6	-4+(-6)=-10

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-4\"$ and $\"-6\"$ add to $\"-10\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-4\"$ and $\"-6\"$ both multiply to $\"24\"$ and add to $\"-10\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-10y\"$ with $\"-4y-6y\"$ . Remember, $\"-4\"$ and $\"-6\"$ add to $\"-10\"$ . So this shows us that $\"-4y-6y=-10y\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"y%5E2%2Bhighlight%28-4y-6y%29%2B24\"$ Replace the second term $\"-10y\"$ with $\"-4y-6y\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%28y%5E2-4y%29%2B%28-6y%2B24%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"y%28y-4%29%2B%28-6y%2B24%29\"$ Factor out the GCF $\"y\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"y%28y-4%29-6%28y-4%29\"$ 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( " $\"%28y-6%29%28y-4%29\"$ Combine like terms. Or factor out the common term $\"y-4\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"-1%28y%5E2-10y%2B24%29\"$ then factors further to $\"-%28y-6%29%28y-4%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"-y%5E2%2B10y-24\"$ completely factors to $\"-%28y-6%29%28y-4%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"-y%5E2%2B10y-24=-%28y-6%29%28y-4%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"-%28y-6%29%28y-4%29\"$ to get $\"-y%5E2%2B10y-24\"$ or by graphing the original expression and the answer (the two graphs should be identical).\r
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