document.write( "Question 683092: factor the trinomial 9a*2+3a-20 \n" ); document.write( "

Algebra.Com's Answer #423468 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"9a%5E2%2B3a-20\"$ , we can see that the first coefficient is $\"9\"$ , the second coefficient is $\"3\"$ , and the last term is $\"-20\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"9\"$ by the last term $\"-20\"$ to get $\"%289%29%28-20%29=-180\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-180\"$ (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 $\"-180\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-180\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,5,6,9,10,12,15,18,20,30,36,45,60,90,180\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-5,-6,-9,-10,-12,-15,-18,-20,-30,-36,-45,-60,-90,-180\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 $\"-180\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-180) = -180
\n" ); document.write( "2*(-90) = -180
\n" ); document.write( "3*(-60) = -180
\n" ); document.write( "4*(-45) = -180
\n" ); document.write( "5*(-36) = -180
\n" ); document.write( "6*(-30) = -180
\n" ); document.write( "9*(-20) = -180
\n" ); document.write( "10*(-18) = -180
\n" ); document.write( "12*(-15) = -180
\n" ); document.write( "(-1)*(180) = -180
\n" ); document.write( "(-2)*(90) = -180
\n" ); document.write( "(-3)*(60) = -180
\n" ); document.write( "(-4)*(45) = -180
\n" ); document.write( "(-5)*(36) = -180
\n" ); document.write( "(-6)*(30) = -180
\n" ); document.write( "(-9)*(20) = -180
\n" ); document.write( "(-10)*(18) = -180
\n" ); document.write( "(-12)*(15) = -180\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	-180	1+(-180)=-179
2	-90	2+(-90)=-88
3	-60	3+(-60)=-57
4	-45	4+(-45)=-41
5	-36	5+(-36)=-31
6	-30	6+(-30)=-24
9	-20	9+(-20)=-11
10	-18	10+(-18)=-8
12	-15	12+(-15)=-3
-1	180	-1+180=179
-2	90	-2+90=88
-3	60	-3+60=57
-4	45	-4+45=41
-5	36	-5+36=31
-6	30	-6+30=24
-9	20	-9+20=11
-10	18	-10+18=8
-12	15	-12+15=3

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-12\"$ and $\"15\"$ add to $\"3\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-12\"$ and $\"15\"$ both multiply to $\"-180\"$ and add to $\"3\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"3a\"$ with $\"-12a%2B15a\"$ . Remember, $\"-12\"$ and $\"15\"$ add to $\"3\"$ . So this shows us that $\"-12a%2B15a=3a\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"9a%5E2%2Bhighlight%28-12a%2B15a%29-20\"$ Replace the second term $\"3a\"$ with $\"-12a%2B15a\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%289a%5E2-12a%29%2B%2815a-20%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"3a%283a-4%29%2B%2815a-20%29\"$ Factor out the GCF $\"3a\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"3a%283a-4%29%2B5%283a-4%29\"$ 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( " $\"%283a%2B5%29%283a-4%29\"$ Combine like terms. Or factor out the common term $\"3a-4\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"9a%5E2%2B3a-20\"$ factors to $\"%283a%2B5%29%283a-4%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"9a%5E2%2B3a-20=%283a%2B5%29%283a-4%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%283a%2B5%29%283a-4%29\"$ to get $\"9a%5E2%2B3a-20\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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