document.write( "Question 638990: x^2+10x+25 \n" ); document.write( "

Algebra.Com's Answer #402504 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"x%5E2%2B10x%2B25\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"10\"$ , and the last term is $\"25\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last term $\"25\"$ to get $\"%281%29%2825%29=25\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"25\"$ (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 $\"25\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"25\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,5,25\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 $\"25\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*25 = 25
\n" ); document.write( "5*5 = 25
\n" ); document.write( "(-1)*(-25) = 25
\n" ); document.write( "(-5)*(-5) = 25\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	25	1+25=26
5	5	5+5=10
-1	-25	-1+(-25)=-26
-5	-5	-5+(-5)=-10

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"5\"$ and $\"5\"$ add to $\"10\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"5\"$ and $\"5\"$ both multiply to $\"25\"$ and add to $\"10\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"10x\"$ with $\"5x%2B5x\"$ . Remember, $\"5\"$ and $\"5\"$ add to $\"10\"$ . So this shows us that $\"5x%2B5x=10x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"x%5E2%2Bhighlight%285x%2B5x%29%2B25\"$ Replace the second term $\"10x\"$ with $\"5x%2B5x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%28x%5E2%2B5x%29%2B%285x%2B25%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x%2B5%29%2B%285x%2B25%29\"$ Factor out the GCF $\"x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x%2B5%29%2B5%28x%2B5%29\"$ Factor out $\"5\"$ 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( " $\"%28x%2B5%29%28x%2B5%29\"$ Combine like terms. Or factor out the common term $\"x%2B5\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"%28x%2B5%29%5E2\"$ 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 $\"x%5E2%2B10x%2B25\"$ factors to $\"%28x%2B5%29%5E2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"x%5E2%2B10x%2B25=%28x%2B5%29%5E2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%28x%2B5%29%5E2\"$ to get $\"x%5E2%2B10x%2B25\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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