document.write( "Question 596104: 20y^2-25y+5 \n" ); document.write( "

Algebra.Com's Answer #377517 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
I'm assuming you want to factor.\r
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\n" ); document.write( "\n" ); document.write( " $\"20y%5E2-25y%2B5\"$ Start with the given expression.\r
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\n" ); document.write( "\n" ); document.write( " $\"5%284y%5E2-5y%2B1%29\"$ Factor out the GCF $\"5\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now let's try to factor the inner expression $\"4y%5E2-5y%2B1\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"4y%5E2-5y%2B1\"$ , we can see that the first coefficient is $\"4\"$ , the second coefficient is $\"-5\"$ , and the last term is $\"1\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"4\"$ by the last term $\"1\"$ to get $\"%284%29%281%29=4\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"4\"$ (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 $\"4\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"4\"$ :\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-4\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 $\"4\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*4 = 4
\n" ); document.write( "2*2 = 4
\n" ); document.write( "(-1)*(-4) = 4
\n" ); document.write( "(-2)*(-2) = 4\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	4	1+4=5
2	2	2+2=4
-1	-4	-1+(-4)=-5
-2	-2	-2+(-2)=-4

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-1\"$ and $\"-4\"$ add to $\"-5\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-1\"$ and $\"-4\"$ both multiply to $\"4\"$ and add to $\"-5\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-5y\"$ with $\"-y-4y\"$ . Remember, $\"-1\"$ and $\"-4\"$ add to $\"-5\"$ . So this shows us that $\"-y-4y=-5y\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"4y%5E2%2Bhighlight%28-y-4y%29%2B1\"$ Replace the second term $\"-5y\"$ with $\"-y-4y\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%284y%5E2-y%29%2B%28-4y%2B1%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"y%284y-1%29%2B%28-4y%2B1%29\"$ Factor out the GCF $\"y\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"y%284y-1%29-1%284y-1%29\"$ Factor out $\"1\"$ 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( " $\"%28y-1%29%284y-1%29\"$ Combine like terms. Or factor out the common term $\"4y-1\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"5%284y%5E2-5y%2B1%29\"$ then factors further to $\"5%28y-1%29%284y-1%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"20y%5E2-25y%2B5\"$ completely factors to $\"5%28y-1%29%284y-1%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"20y%5E2-25y%2B5=5%28y-1%29%284y-1%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"5%28y-1%29%284y-1%29\"$ to get $\"20y%5E2-25y%2B5\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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