document.write( "Question 882098: 9x² + 49x + 20
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Algebra.Com's Answer #532688 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
I'm assuming you want to factor.\r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"9x%5E2%2B49x%2B20\"$ , we can see that the first coefficient is $\"9\"$ , the second coefficient is $\"49\"$ , 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 $\"%289%29%2820%29=180\"$ .\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 $\"49\"$ ?\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 $\"49\"$ :\r
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First Number	Second Number	Sum
1	180	1+180=181
2	90	2+90=92
3	60	3+60=63
4	45	4+45=49
5	36	5+36=41
6	30	6+30=36
9	20	9+20=29
10	18	10+18=28
12	15	12+15=27
-1	-180	-1+(-180)=-181
-2	-90	-2+(-90)=-92
-3	-60	-3+(-60)=-63
-4	-45	-4+(-45)=-49
-5	-36	-5+(-36)=-41
-6	-30	-6+(-30)=-36
-9	-20	-9+(-20)=-29
-10	-18	-10+(-18)=-28
-12	-15	-12+(-15)=-27

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"4\"$ and $\"45\"$ add to $\"49\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"4\"$ and $\"45\"$ both multiply to $\"180\"$ and add to $\"49\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"49x\"$ with $\"4x%2B45x\"$ . Remember, $\"4\"$ and $\"45\"$ add to $\"49\"$ . So this shows us that $\"4x%2B45x=49x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"9x%5E2%2Bhighlight%284x%2B45x%29%2B20\"$ Replace the second term $\"49x\"$ with $\"4x%2B45x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%289x%5E2%2B4x%29%2B%2845x%2B20%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%289x%2B4%29%2B%2845x%2B20%29\"$ Factor out the GCF $\"x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%289x%2B4%29%2B5%289x%2B4%29\"$ 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( " $\"%28x%2B5%29%289x%2B4%29\"$ Combine like terms. Or factor out the common term $\"9x%2B4\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"9x%5E2%2B49x%2B20\"$ factors to $\"%28x%2B5%29%289x%2B4%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"9x%5E2%2B49x%2B20=%28x%2B5%29%289x%2B4%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%28x%2B5%29%289x%2B4%29\"$ to get $\"9x%5E2%2B49x%2B20\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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