document.write( "Question 193222: how do you factor this problem?\r
\n" ); document.write( "\n" ); document.write( "2a^2 + 7a - 4. \n" ); document.write( "

Algebra.Com's Answer #145054 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"2a%5E2%2B7a-4\"$ , we can see that the first coefficient is $\"2\"$ , the second coefficient is $\"7\"$ , and the last term is $\"-4\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"2\"$ by the last term $\"-4\"$ to get $\"%282%29%28-4%29=-8\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-8\"$ (the previous product) and add to the second coefficient $\"7\"$ ?\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 $\"-8\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-8\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,4,8\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-4,-8\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 $\"-8\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-8)
\n" ); document.write( "2*(-4)
\n" ); document.write( "(-1)*(8)
\n" ); document.write( "(-2)*(4)\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 $\"7\"$ :\r
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First Number	Second Number	Sum
1	-8	1+(-8)=-7
2	-4	2+(-4)=-2
-1	8	-1+8=7
-2	4	-2+4=2

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-1\"$ and $\"8\"$ add to $\"7\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-1\"$ and $\"8\"$ both multiply to $\"-8\"$ and add to $\"7\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"7a\"$ with $\"-a%2B8a\"$ . Remember, $\"-1\"$ and $\"8\"$ add to $\"7\"$ . So this shows us that $\"-a%2B8a=7a\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"2a%5E2%2Bhighlight%28-a%2B8a%29-4\"$ Replace the second term $\"7a\"$ with $\"-a%2B8a\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%282a%5E2-a%29%2B%288a-4%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"a%282a-1%29%2B%288a-4%29\"$ Factor out the GCF $\"a\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"a%282a-1%29%2B4%282a-1%29\"$ Factor out $\"4\"$ 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( " $\"%28a%2B4%29%282a-1%29\"$ Combine like terms. Or factor out the common term $\"2a-1\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"2a%5E2%2B7a-4\"$ factors to $\"%28a%2B4%29%282a-1%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by FOILing $\"%28a%2B4%29%282a-1%29\"$ to get $\"2a%5E2%2B7a-4\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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