document.write( "Question 581995: Factor.
\n" ); document.write( "s^2+14s+49 \n" ); document.write( "

Algebra.Com's Answer #371904 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"s%5E2%2B14s%2B49\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"14\"$ , and the last term is $\"49\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last term $\"49\"$ to get $\"%281%29%2849%29=49\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"49\"$ (the previous product) and add to the second coefficient $\"14\"$ ?\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 $\"49\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"49\"$ :\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 $\"49\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*49 = 49
\n" ); document.write( "7*7 = 49
\n" ); document.write( "(-1)*(-49) = 49
\n" ); document.write( "(-7)*(-7) = 49\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 $\"14\"$ :\r
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First Number	Second Number	Sum
1	49	1+49=50
7	7	7+7=14
-1	-49	-1+(-49)=-50
-7	-7	-7+(-7)=-14

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"7\"$ and $\"7\"$ add to $\"14\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"7\"$ and $\"7\"$ both multiply to $\"49\"$ and add to $\"14\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"14s\"$ with $\"7s%2B7s\"$ . Remember, $\"7\"$ and $\"7\"$ add to $\"14\"$ . So this shows us that $\"7s%2B7s=14s\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"s%5E2%2Bhighlight%287s%2B7s%29%2B49\"$ Replace the second term $\"14s\"$ with $\"7s%2B7s\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%28s%5E2%2B7s%29%2B%287s%2B49%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"s%28s%2B7%29%2B%287s%2B49%29\"$ Factor out the GCF $\"s\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"s%28s%2B7%29%2B7%28s%2B7%29\"$ Factor out $\"7\"$ 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( " $\"%28s%2B7%29%28s%2B7%29\"$ Combine like terms. Or factor out the common term $\"s%2B7\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"%28s%2B7%29%5E2\"$ 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 $\"s%5E2%2B14s%2B49\"$ factors to $\"%28s%2B7%29%5E2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"s%5E2%2B14s%2B49=%28s%2B7%29%5E2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%28s%2B7%29%5E2\"$ to get $\"s%5E2%2B14s%2B49\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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