document.write( "Question 604356: Can 5x^2 + 2x - 13 = 0 be factored? \n" ); document.write( "

Algebra.Com's Answer #381116 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"5x%5E2%2B2x-13\"$ , we can see that the first coefficient is $\"5\"$ , the second coefficient is $\"2\"$ , and the last term is $\"-13\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"5\"$ by the last term $\"-13\"$ to get $\"%285%29%28-13%29=-65\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-65\"$ (the previous product) and add to the second coefficient $\"2\"$ ?\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 $\"-65\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-65\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,5,13,65\r
\n" ); document.write( "\n" ); document.write( "-1,-5,-13,-65\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 $\"-65\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-65) = -65
\n" ); document.write( "5*(-13) = -65
\n" ); document.write( "(-1)*(65) = -65
\n" ); document.write( "(-5)*(13) = -65\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 $\"2\"$ :\r
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First Number	Second Number	Sum
1	-65	1+(-65)=-64
5	-13	5+(-13)=-8
-1	65	-1+65=64
-5	13	-5+13=8

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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 $\"2\"$ . So $\"5x%5E2%2B2x-13\"$ cannot be factored.\r
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\n" ); document.write( "\n" ); document.write( "This tells us that $\"5x%5E2%2B2x-13\"$ is prime.\r
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\n" ); document.write( "\n" ); document.write( "This means that you'll have to either complete the square or use the quadratic formula to solve the given equation. \n" ); document.write( "

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