document.write( "Question 601423: Answer to 6a^2-a-35 \n" ); document.write( "

Algebra.Com's Answer #379747 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"6a%5E2-a-35\"$ , we can see that the first coefficient is $\"6\"$ , the second coefficient is $\"-1\"$ , and the last term is $\"-35\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"6\"$ by the last term $\"-35\"$ to get $\"%286%29%28-35%29=-210\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-210\"$ (the previous product) and add to the second coefficient $\"-1\"$ ?\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 $\"-210\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-210\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,5,6,7,10,14,15,21,30,35,42,70,105,210\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-5,-6,-7,-10,-14,-15,-21,-30,-35,-42,-70,-105,-210\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 $\"-210\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-210) = -210
\n" ); document.write( "2*(-105) = -210
\n" ); document.write( "3*(-70) = -210
\n" ); document.write( "5*(-42) = -210
\n" ); document.write( "6*(-35) = -210
\n" ); document.write( "7*(-30) = -210
\n" ); document.write( "10*(-21) = -210
\n" ); document.write( "14*(-15) = -210
\n" ); document.write( "(-1)*(210) = -210
\n" ); document.write( "(-2)*(105) = -210
\n" ); document.write( "(-3)*(70) = -210
\n" ); document.write( "(-5)*(42) = -210
\n" ); document.write( "(-6)*(35) = -210
\n" ); document.write( "(-7)*(30) = -210
\n" ); document.write( "(-10)*(21) = -210
\n" ); document.write( "(-14)*(15) = -210\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 $\"-1\"$ :\r
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First Number	Second Number	Sum
1	-210	1+(-210)=-209
2	-105	2+(-105)=-103
3	-70	3+(-70)=-67
5	-42	5+(-42)=-37
6	-35	6+(-35)=-29
7	-30	7+(-30)=-23
10	-21	10+(-21)=-11
14	-15	14+(-15)=-1
-1	210	-1+210=209
-2	105	-2+105=103
-3	70	-3+70=67
-5	42	-5+42=37
-6	35	-6+35=29
-7	30	-7+30=23
-10	21	-10+21=11
-14	15	-14+15=1

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"14\"$ and $\"-15\"$ add to $\"-1\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"14\"$ and $\"-15\"$ both multiply to $\"-210\"$ and add to $\"-1\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-1a\"$ with $\"14a-15a\"$ . Remember, $\"14\"$ and $\"-15\"$ add to $\"-1\"$ . So this shows us that $\"14a-15a=-1a\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"6a%5E2%2Bhighlight%2814a-15a%29-35\"$ Replace the second term $\"-1a\"$ with $\"14a-15a\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%286a%5E2%2B14a%29%2B%28-15a-35%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"2a%283a%2B7%29%2B%28-15a-35%29\"$ Factor out the GCF $\"2a\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"2a%283a%2B7%29-5%283a%2B7%29\"$ 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( " $\"%282a-5%29%283a%2B7%29\"$ Combine like terms. Or factor out the common term $\"3a%2B7\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"6a%5E2-a-35\"$ factors to $\"%282a-5%29%283a%2B7%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"6a%5E2-a-35=%282a-5%29%283a%2B7%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%282a-5%29%283a%2B7%29\"$ to get $\"6a%5E2-a-35\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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