document.write( "Question 282083: Factor completely. Show all work neccessary.\r
\n" ); document.write( "\n" ); document.write( "x^2 - 14x + 49 \n" ); document.write( "

Algebra.Com's Answer #205383 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"x%5E2-14x%2B49\"$ , 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
\n" ); document.write( "\n" ); document.write( "1,7,49\r
\n" ); document.write( "\n" ); document.write( "-1,-7,-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 $\"-14x\"$ with $\"-7x-7x\"$ . Remember, $\"-7\"$ and $\"-7\"$ add to $\"-14\"$ . So this shows us that $\"-7x-7x=-14x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"x%5E2%2Bhighlight%28-7x-7x%29%2B49\"$ Replace the second term $\"-14x\"$ with $\"-7x-7x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%28x%5E2-7x%29%2B%28-7x%2B49%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x-7%29%2B%28-7x%2B49%29\"$ Factor out the GCF $\"x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x-7%29-7%28x-7%29\"$ 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( " $\"%28x-7%29%28x-7%29\"$ Combine like terms. Or factor out the common term $\"x-7\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"%28x-7%29%5E2\"$ 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 $\"x%5E2-14x%2B49\"$ factors to $\"%28x-7%29%5E2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"x%5E2-14x%2B49=%28x-7%29%5E2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%28x-7%29%5E2\"$ to get $\"x%5E2-14x%2B49\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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