document.write( "Question 383973: Factor the quadratic expression.\r
\n" ); document.write( "\n" ); document.write( "y^2 - 4y + 4 \n" ); document.write( "

Algebra.Com's Answer #271848 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"y%5E2-4y%2B4\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"-4\"$ , and the last term is $\"4\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last term $\"4\"$ to get $\"%281%29%284%29=4\"$ .\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 $\"-4\"$ ?\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 $\"-4\"$ :\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 $\"-2\"$ and $\"-2\"$ add to $\"-4\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-2\"$ and $\"-2\"$ both multiply to $\"4\"$ and add to $\"-4\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-4y\"$ with $\"-2y-2y\"$ . Remember, $\"-2\"$ and $\"-2\"$ add to $\"-4\"$ . So this shows us that $\"-2y-2y=-4y\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"y%5E2%2Bhighlight%28-2y-2y%29%2B4\"$ Replace the second term $\"-4y\"$ with $\"-2y-2y\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%28y%5E2-2y%29%2B%28-2y%2B4%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"y%28y-2%29%2B%28-2y%2B4%29\"$ Factor out the GCF $\"y\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"y%28y-2%29-2%28y-2%29\"$ 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( " $\"%28y-2%29%28y-2%29\"$ Combine like terms. Or factor out the common term $\"y-2\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"%28y-2%29%5E2\"$ Condense the terms.\r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"y%5E2-4y%2B4\"$ factors to $\"%28y-2%29%5E2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"y%5E2-4y%2B4=%28y-2%29%5E2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%28y-2%29%5E2\"$ to get $\"y%5E2-4y%2B4\"$ 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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