document.write( "Question 317097: I am having trouble figuring out how to factor the three problems listed below:1. x^2+2x-2x-4;2.v^2+9v+14;3. r^2-2r-48. I have been working on them for about 1 1/2 hours at this time and just can not seem to get any where can you help? \n" ); document.write( "

Algebra.Com's Answer #227020 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
# 1\r
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\n" ); document.write( "\n" ); document.write( " $\"x%5E2%2B2x-2x-4\"$ Start with the given expression\r
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\n" ); document.write( "\n" ); document.write( " $\"%28x%5E2%2B2x%29%2B%28-2x-4%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x%2B2%29-2%28x%2B2%29\"$ Factor out the GCF $\"x\"$ out of the first group. Factor out the GCF $\"-2\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%28x-2%29%28x%2B2%29\"$ Since we have the common term $\"x%2B2\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"x%5E2%2B2x-2x-4\"$ factors to $\"%28x-2%29%28x%2B2%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"x%5E2%2B2x-2x-4=%28x-2%29%28x%2B2%29\"$ \r
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\n" ); document.write( "# 2\r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"v%5E2%2B9v%2B14\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"9\"$ , and the last term is $\"14\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last term $\"14\"$ to get $\"%281%29%2814%29=14\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"14\"$ (the previous product) and add to the second coefficient $\"9\"$ ?\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 $\"14\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"14\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,7,14\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-7,-14\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 $\"14\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*14 = 14
\n" ); document.write( "2*7 = 14
\n" ); document.write( "(-1)*(-14) = 14
\n" ); document.write( "(-2)*(-7) = 14\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 $\"9\"$ :\r
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First Number	Second Number	Sum
1	14	1+14=15
2	7	2+7=9
-1	-14	-1+(-14)=-15
-2	-7	-2+(-7)=-9

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"2\"$ and $\"7\"$ add to $\"9\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"2\"$ and $\"7\"$ both multiply to $\"14\"$ and add to $\"9\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"9v\"$ with $\"2v%2B7v\"$ . Remember, $\"2\"$ and $\"7\"$ add to $\"9\"$ . So this shows us that $\"2v%2B7v=9v\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"v%5E2%2Bhighlight%282v%2B7v%29%2B14\"$ Replace the second term $\"9v\"$ with $\"2v%2B7v\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%28v%5E2%2B2v%29%2B%287v%2B14%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"v%28v%2B2%29%2B%287v%2B14%29\"$ Factor out the GCF $\"v\"$ from the first group.\r
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\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"v%28v%2B2%29%2B7%28v%2B2%29\"$ 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( " $\"%28v%2B7%29%28v%2B2%29\"$ Combine like terms. Or factor out the common term $\"v%2B2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"v%5E2%2B9v%2B14\"$ factors to $\"%28v%2B7%29%28v%2B2%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"v%5E2%2B9v%2B14=%28v%2B7%29%28v%2B2%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%28v%2B7%29%28v%2B2%29\"$ to get $\"v%5E2%2B9v%2B14\"$ or by graphing the original expression and the answer (the two graphs should be identical).\r
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\n" ); document.write( "# 3\r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"r%5E2-2r-48\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"-2\"$ , and the last term is $\"-48\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last term $\"-48\"$ to get $\"%281%29%28-48%29=-48\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-48\"$ (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 $\"-48\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-48\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,8,12,16,24,48\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-8,-12,-16,-24,-48\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Note: list the negative of each factor. This will allow us to find all possible combinations.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "These factors pair up and multiply to $\"-48\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-48) = -48
\n" ); document.write( "2*(-24) = -48
\n" ); document.write( "3*(-16) = -48
\n" ); document.write( "4*(-12) = -48
\n" ); document.write( "6*(-8) = -48
\n" ); document.write( "(-1)*(48) = -48
\n" ); document.write( "(-2)*(24) = -48
\n" ); document.write( "(-3)*(16) = -48
\n" ); document.write( "(-4)*(12) = -48
\n" ); document.write( "(-6)*(8) = -48\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	-48	1+(-48)=-47
2	-24	2+(-24)=-22
3	-16	3+(-16)=-13
4	-12	4+(-12)=-8
6	-8	6+(-8)=-2
-1	48	-1+48=47
-2	24	-2+24=22
-3	16	-3+16=13
-4	12	-4+12=8
-6	8	-6+8=2

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"6\"$ and $\"-8\"$ add to $\"-2\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"6\"$ and $\"-8\"$ both multiply to $\"-48\"$ and add to $\"-2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-2r\"$ with $\"6r-8r\"$ . Remember, $\"6\"$ and $\"-8\"$ add to $\"-2\"$ . So this shows us that $\"6r-8r=-2r\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"r%5E2%2Bhighlight%286r-8r%29-48\"$ Replace the second term $\"-2r\"$ with $\"6r-8r\"$ .\r
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\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%28r%5E2%2B6r%29%2B%28-8r-48%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"r%28r%2B6%29%2B%28-8r-48%29\"$ Factor out the GCF $\"r\"$ from the first group.\r
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\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"r%28r%2B6%29-8%28r%2B6%29\"$ Factor out $\"8\"$ 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( "\n" ); document.write( " $\"%28r-8%29%28r%2B6%29\"$ Combine like terms. Or factor out the common term $\"r%2B6\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"r%5E2-2r-48\"$ factors to $\"%28r-8%29%28r%2B6%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"r%5E2-2r-48=%28r-8%29%28r%2B6%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%28r-8%29%28r%2B6%29\"$ to get $\"r%5E2-2r-48\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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