document.write( "Question 549824: factor x^2+36x+308 \n" ); document.write( "

Algebra.Com's Answer #358097 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"x%5E2%2B36x%2B308\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"36\"$ , and the last term is $\"308\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last term $\"308\"$ to get $\"%281%29%28308%29=308\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"308\"$ (the previous product) and add to the second coefficient $\"36\"$ ?\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 $\"308\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"308\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,4,7,11,14,22,28,44,77,154,308\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-4,-7,-11,-14,-22,-28,-44,-77,-154,-308\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 $\"308\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*308 = 308
\n" ); document.write( "2*154 = 308
\n" ); document.write( "4*77 = 308
\n" ); document.write( "7*44 = 308
\n" ); document.write( "11*28 = 308
\n" ); document.write( "14*22 = 308
\n" ); document.write( "(-1)*(-308) = 308
\n" ); document.write( "(-2)*(-154) = 308
\n" ); document.write( "(-4)*(-77) = 308
\n" ); document.write( "(-7)*(-44) = 308
\n" ); document.write( "(-11)*(-28) = 308
\n" ); document.write( "(-14)*(-22) = 308\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 $\"36\"$ :\r
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First Number	Second Number	Sum
1	308	1+308=309
2	154	2+154=156
4	77	4+77=81
7	44	7+44=51
11	28	11+28=39
14	22	14+22=36
-1	-308	-1+(-308)=-309
-2	-154	-2+(-154)=-156
-4	-77	-4+(-77)=-81
-7	-44	-7+(-44)=-51
-11	-28	-11+(-28)=-39
-14	-22	-14+(-22)=-36

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"14\"$ and $\"22\"$ add to $\"36\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"14\"$ and $\"22\"$ both multiply to $\"308\"$ and add to $\"36\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"36x\"$ with $\"14x%2B22x\"$ . Remember, $\"14\"$ and $\"22\"$ add to $\"36\"$ . So this shows us that $\"14x%2B22x=36x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"x%5E2%2Bhighlight%2814x%2B22x%29%2B308\"$ Replace the second term $\"36x\"$ with $\"14x%2B22x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%28x%5E2%2B14x%29%2B%2822x%2B308%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x%2B14%29%2B%2822x%2B308%29\"$ Factor out the GCF $\"x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x%2B14%29%2B22%28x%2B14%29\"$ Factor out $\"22\"$ 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( " $\"%28x%2B22%29%28x%2B14%29\"$ Combine like terms. Or factor out the common term $\"x%2B14\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"x%5E2%2B36x%2B308\"$ factors to $\"%28x%2B22%29%28x%2B14%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"x%5E2%2B36x%2B308=%28x%2B22%29%28x%2B14%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%28x%2B22%29%28x%2B14%29\"$ to get $\"x%5E2%2B36x%2B308\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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