document.write( "Question 207720This question is from textbook
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\n" ); document.write( "a^2+20a+100 (?)^2 \n" ); document.write( "

Algebra.Com's Answer #157109 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"a%5E2%2B20a%2B100\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"20\"$ , and the last term is $\"100\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last term $\"100\"$ to get $\"%281%29%28100%29=100\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"100\"$ (the previous product) and add to the second coefficient $\"20\"$ ?\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 $\"100\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"100\"$ :\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-4,-5,-10,-20,-25,-50,-100\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 $\"100\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*100 = 100
\n" ); document.write( "2*50 = 100
\n" ); document.write( "4*25 = 100
\n" ); document.write( "5*20 = 100
\n" ); document.write( "10*10 = 100
\n" ); document.write( "(-1)*(-100) = 100
\n" ); document.write( "(-2)*(-50) = 100
\n" ); document.write( "(-4)*(-25) = 100
\n" ); document.write( "(-5)*(-20) = 100
\n" ); document.write( "(-10)*(-10) = 100\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 $\"20\"$ :\r
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First Number	Second Number	Sum
1	100	1+100=101
2	50	2+50=52
4	25	4+25=29
5	20	5+20=25
10	10	10+10=20
-1	-100	-1+(-100)=-101
-2	-50	-2+(-50)=-52
-4	-25	-4+(-25)=-29
-5	-20	-5+(-20)=-25
-10	-10	-10+(-10)=-20

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"10\"$ and $\"10\"$ add to $\"20\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"10\"$ and $\"10\"$ both multiply to $\"100\"$ and add to $\"20\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"20a\"$ with $\"10a%2B10a\"$ . Remember, $\"10\"$ and $\"10\"$ add to $\"20\"$ . So this shows us that $\"10a%2B10a=20a\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"a%5E2%2Bhighlight%2810a%2B10a%29%2B100\"$ Replace the second term $\"20a\"$ with $\"10a%2B10a\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%28a%5E2%2B10a%29%2B%2810a%2B100%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"a%28a%2B10%29%2B%2810a%2B100%29\"$ Factor out the GCF $\"a\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"a%28a%2B10%29%2B10%28a%2B10%29\"$ Factor out $\"10\"$ 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( " $\"%28a%2B10%29%28a%2B10%29\"$ Combine like terms. Or factor out the common term $\"a%2B10\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"%28a%2B10%29%5E2\"$ 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 $\"a%5E2%2B20a%2B100\"$ factors to $\"%28a%2B10%29%5E2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"a%5E2%2B20a%2B100=%28a%2B10%29%5E2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%28a%2B10%29%5E2\"$ to get $\"a%5E2%2B20a%2B100\"$ or by graphing the original expression and the answer (the two graphs should be identical).\r
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