document.write( "Question 183250: w^2+14w+49 \n" ); document.write( "

Algebra.Com's Answer #137584 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
I'm assuming that you want to factor this.\r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"w%5E2%2B14w%2B49\"$ , 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( "-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
\n" ); document.write( "7*7
\n" ); document.write( "(-1)*(-49)
\n" ); document.write( "(-7)*(-7)\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 $\"14w\"$ with $\"7w%2B7w\"$ . Remember, $\"7\"$ and $\"7\"$ add to $\"14\"$ . So this shows us that $\"7w%2B7w=14w\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"w%5E2%2Bhighlight%287w%2B7w%29%2B49\"$ Replace the second term $\"14w\"$ with $\"7w%2B7w\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%28w%5E2%2B7w%29%2B%287w%2B49%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"w%28w%2B7%29%2B%287w%2B49%29\"$ Factor out the GCF $\"w\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"w%28w%2B7%29%2B7%28w%2B7%29\"$ 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( " $\"%28w%2B7%29%28w%2B7%29\"$ Combine like terms. Or factor out the common term $\"w%2B7\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"%28w%2B7%29%5E2\"$ Condense the factors\r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"w%5E2%2B14w%2B49\"$ factors to $\"%28w%2B7%29%5E2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"w%5E2%2B14w%2B49=%28w%2B7%29%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by FOILing $\"%28w%2B7%29%28w%2B7%29\"$ to get $\"w%5E2%2B14w%2B49\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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