document.write( "Question 320559: help me solve equation (3x^2+49x+72) in form (3x+ ) (x+ ) \n" ); document.write( "

Algebra.Com's Answer #229566 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"3x%5E2%2B49x%2B72\"$ , we can see that the first coefficient is $\"3\"$ , the second coefficient is $\"49\"$ , and the last term is $\"72\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"3\"$ by the last term $\"72\"$ to get $\"%283%29%2872%29=216\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"216\"$ (the previous product) and add to the second coefficient $\"49\"$ ?\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 $\"216\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"216\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,8,9,12,18,24,27,36,54,72,108,216\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-8,-9,-12,-18,-24,-27,-36,-54,-72,-108,-216\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 $\"216\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*216 = 216
\n" ); document.write( "2*108 = 216
\n" ); document.write( "3*72 = 216
\n" ); document.write( "4*54 = 216
\n" ); document.write( "6*36 = 216
\n" ); document.write( "8*27 = 216
\n" ); document.write( "9*24 = 216
\n" ); document.write( "12*18 = 216
\n" ); document.write( "(-1)*(-216) = 216
\n" ); document.write( "(-2)*(-108) = 216
\n" ); document.write( "(-3)*(-72) = 216
\n" ); document.write( "(-4)*(-54) = 216
\n" ); document.write( "(-6)*(-36) = 216
\n" ); document.write( "(-8)*(-27) = 216
\n" ); document.write( "(-9)*(-24) = 216
\n" ); document.write( "(-12)*(-18) = 216\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 $\"49\"$ :\r
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First Number	Second Number	Sum
1	216	1+216=217
2	108	2+108=110
3	72	3+72=75
4	54	4+54=58
6	36	6+36=42
8	27	8+27=35
9	24	9+24=33
12	18	12+18=30
-1	-216	-1+(-216)=-217
-2	-108	-2+(-108)=-110
-3	-72	-3+(-72)=-75
-4	-54	-4+(-54)=-58
-6	-36	-6+(-36)=-42
-8	-27	-8+(-27)=-35
-9	-24	-9+(-24)=-33
-12	-18	-12+(-18)=-30

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that there are no pairs of numbers which add to $\"49\"$ . So $\"3x%5E2%2B49x%2B72\"$ cannot be factored.\r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"3x%5E2%2B49x%2B72\"$ doesn't factor at all (over the rational numbers).\r
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\n" ); document.write( "\n" ); document.write( "So $\"3x%5E2%2B49x%2B72\"$ is prime.\r
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\n" ); document.write( "\n" ); document.write( "Make sure that you have the correct problem.
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