document.write( "Question 191767: Factor \r
\n" ); document.write( "\n" ); document.write( "2x^2+3x-5 \n" ); document.write( "

Algebra.Com's Answer #143887 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"2x%5E2%2B3x-5\"$ , we can see that the first coefficient is $\"2\"$ , the second coefficient is $\"3\"$ , and the last term is $\"-5\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"2\"$ by the last term $\"-5\"$ to get $\"%282%29%28-5%29=-10\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-10\"$ (the previous product) and add to the second coefficient $\"3\"$ ?\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 $\"-10\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-10\"$ :\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-5,-10\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 $\"-10\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-10)
\n" ); document.write( "2*(-5)
\n" ); document.write( "(-1)*(10)
\n" ); document.write( "(-2)*(5)\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 $\"3\"$ :\r
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First Number	Second Number	Sum
1	-10	1+(-10)=-9
2	-5	2+(-5)=-3
-1	10	-1+10=9
-2	5	-2+5=3

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-2\"$ and $\"5\"$ add to $\"3\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-2\"$ and $\"5\"$ both multiply to $\"-10\"$ and add to $\"3\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"3x\"$ with $\"-2x%2B5x\"$ . Remember, $\"-2\"$ and $\"5\"$ add to $\"3\"$ . So this shows us that $\"-2x%2B5x=3x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"2x%5E2%2Bhighlight%28-2x%2B5x%29-5\"$ Replace the second term $\"3x\"$ with $\"-2x%2B5x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%282x%5E2-2x%29%2B%285x-5%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"2x%28x-1%29%2B%285x-5%29\"$ Factor out the GCF $\"2x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"2x%28x-1%29%2B5%28x-1%29\"$ Factor out $\"5\"$ 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( " $\"%282x%2B5%29%28x-1%29\"$ Combine like terms. Or factor out the common term $\"x-1\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"2x%5E2%2B3x-5\"$ factors to $\"%282x%2B5%29%28x-1%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by FOILing $\"%282x%2B5%29%28x-1%29\"$ to get $\"2x%5E2%2B3x-5\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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