document.write( "Question 299373: reverse foil or grouping
\n" ); document.write( "2x^2+15x+25 \n" ); document.write( "

Algebra.Com's Answer #215143 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"2x%5E2%2B15x%2B25\"$ , we can see that the first coefficient is $\"2\"$ , the second coefficient is $\"15\"$ , and the last term is $\"25\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"2\"$ by the last term $\"25\"$ to get $\"%282%29%2825%29=50\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"50\"$ (the previous product) and add to the second coefficient $\"15\"$ ?\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 $\"50\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"50\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,5,10,25,50\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-5,-10,-25,-50\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 $\"50\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*50 = 50
\n" ); document.write( "2*25 = 50
\n" ); document.write( "5*10 = 50
\n" ); document.write( "(-1)*(-50) = 50
\n" ); document.write( "(-2)*(-25) = 50
\n" ); document.write( "(-5)*(-10) = 50\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 $\"15\"$ :\r
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First Number	Second Number	Sum
1	50	1+50=51
2	25	2+25=27
5	10	5+10=15
-1	-50	-1+(-50)=-51
-2	-25	-2+(-25)=-27
-5	-10	-5+(-10)=-15

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"5\"$ and $\"10\"$ add to $\"15\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"5\"$ and $\"10\"$ both multiply to $\"50\"$ and add to $\"15\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"15x\"$ with $\"5x%2B10x\"$ . Remember, $\"5\"$ and $\"10\"$ add to $\"15\"$ . So this shows us that $\"5x%2B10x=15x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"2x%5E2%2Bhighlight%285x%2B10x%29%2B25\"$ Replace the second term $\"15x\"$ with $\"5x%2B10x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%282x%5E2%2B5x%29%2B%2810x%2B25%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%282x%2B5%29%2B%2810x%2B25%29\"$ Factor out the GCF $\"x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%282x%2B5%29%2B5%282x%2B5%29\"$ 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( " $\"%28x%2B5%29%282x%2B5%29\"$ Combine like terms. Or factor out the common term $\"2x%2B5\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"2x%5E2%2B15x%2B25\"$ factors to $\"%28x%2B5%29%282x%2B5%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"2x%5E2%2B15x%2B25=%28x%2B5%29%282x%2B5%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%28x%2B5%29%282x%2B5%29\"$ to get $\"2x%5E2%2B15x%2B25\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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