document.write( "Question 199751: I need to use factoring to solve an equation. I know the procedure but need to review one thing:
\n" ); document.write( "Solve: -2x^2 + 13x -21 =0
\n" ); document.write( "I know this becomes:
\n" ); document.write( " 2x^2 -13x +21 + 0\r
\n" ); document.write( "\n" ); document.write( " (2x ) (x )
\n" ); document.write( "what I need is what goes in each parenthenes. the correct products of -21 should equal the sum of -13. I only see 21 x -1, -21 x 1, 7 x -3, and -7 x 3. But none of these add up to -13. \n" ); document.write( "

Algebra.Com's Answer #150140 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
Here's one way you can factor $\"+2x%5E2+-13x+%2B21\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"2x%5E2-13x%2B21\"$ , we can see that the first coefficient is $\"2\"$ , the second coefficient is $\"-13\"$ , and the last term is $\"21\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"2\"$ by the last term $\"21\"$ to get $\"%282%29%2821%29=42\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"42\"$ (the previous product) and add to the second coefficient $\"-13\"$ ?\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 $\"42\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"42\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,6,7,14,21,42\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-6,-7,-14,-21,-42\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 $\"42\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*42
\n" ); document.write( "2*21
\n" ); document.write( "3*14
\n" ); document.write( "6*7
\n" ); document.write( "(-1)*(-42)
\n" ); document.write( "(-2)*(-21)
\n" ); document.write( "(-3)*(-14)
\n" ); document.write( "(-6)*(-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 $\"-13\"$ :\r
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First Number	Second Number	Sum
1	42	1+42=43
2	21	2+21=23
3	14	3+14=17
6	7	6+7=13
-1	-42	-1+(-42)=-43
-2	-21	-2+(-21)=-23
-3	-14	-3+(-14)=-17
-6	-7	-6+(-7)=-13

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-6\"$ and $\"-7\"$ add to $\"-13\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-6\"$ and $\"-7\"$ both multiply to $\"42\"$ and add to $\"-13\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-13x\"$ with $\"-6x-7x\"$ . Remember, $\"-6\"$ and $\"-7\"$ add to $\"-13\"$ . So this shows us that $\"-6x-7x=-13x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"2x%5E2%2Bhighlight%28-6x-7x%29%2B21\"$ Replace the second term $\"-13x\"$ with $\"-6x-7x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%282x%5E2-6x%29%2B%28-7x%2B21%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"2x%28x-3%29%2B%28-7x%2B21%29\"$ Factor out the GCF $\"2x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"2x%28x-3%29-7%28x-3%29\"$ 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( " $\"%282x-7%29%28x-3%29\"$ Combine like terms. Or factor out the common term $\"x-3\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"2x%5E2-13x%2B21\"$ factors to $\"%282x-7%29%28x-3%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by FOILing $\"%282x-7%29%28x-3%29\"$ to get $\"2x%5E2-13x%2B21\"$ or by graphing the original expression and the answer (the two graphs should be identical).\r
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\n" ); document.write( "\n" ); document.write( "If you have any questions, email me at jim_thompson5910@hotmail.com.
\n" ); document.write( "Check out my website if you are interested in tutoring. \n" ); document.write( "

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