document.write( "Question 199600: I'm not to sure if this is the right section for my question but here goes:
\n" ); document.write( "I'm working on factorizing quadratic trinomials and here is a question I need help with factorizing this expression.
\n" ); document.write( "1) 20x^2-44x+21 \n" ); document.write( "

Algebra.Com's Answer #150015 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"20x%5E2-44x%2B21\"$ , we can see that the first coefficient is $\"20\"$ , the second coefficient is $\"-44\"$ , and the last term is $\"21\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"20\"$ by the last term $\"21\"$ to get $\"%2820%29%2821%29=420\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"420\"$ (the previous product) and add to the second coefficient $\"-44\"$ ?\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 $\"420\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"420\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,5,6,7,10,12,14,15,20,21,28,30,35,42,60,70,84,105,140,210,420\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-5,-6,-7,-10,-12,-14,-15,-20,-21,-28,-30,-35,-42,-60,-70,-84,-105,-140,-210,-420\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 $\"420\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*420
\n" ); document.write( "2*210
\n" ); document.write( "3*140
\n" ); document.write( "4*105
\n" ); document.write( "5*84
\n" ); document.write( "6*70
\n" ); document.write( "7*60
\n" ); document.write( "10*42
\n" ); document.write( "12*35
\n" ); document.write( "14*30
\n" ); document.write( "15*28
\n" ); document.write( "20*21
\n" ); document.write( "(-1)*(-420)
\n" ); document.write( "(-2)*(-210)
\n" ); document.write( "(-3)*(-140)
\n" ); document.write( "(-4)*(-105)
\n" ); document.write( "(-5)*(-84)
\n" ); document.write( "(-6)*(-70)
\n" ); document.write( "(-7)*(-60)
\n" ); document.write( "(-10)*(-42)
\n" ); document.write( "(-12)*(-35)
\n" ); document.write( "(-14)*(-30)
\n" ); document.write( "(-15)*(-28)
\n" ); document.write( "(-20)*(-21)\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 $\"-44\"$ :\r
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First Number	Second Number	Sum
1	420	1+420=421
2	210	2+210=212
3	140	3+140=143
4	105	4+105=109
5	84	5+84=89
6	70	6+70=76
7	60	7+60=67
10	42	10+42=52
12	35	12+35=47
14	30	14+30=44
15	28	15+28=43
20	21	20+21=41
-1	-420	-1+(-420)=-421
-2	-210	-2+(-210)=-212
-3	-140	-3+(-140)=-143
-4	-105	-4+(-105)=-109
-5	-84	-5+(-84)=-89
-6	-70	-6+(-70)=-76
-7	-60	-7+(-60)=-67
-10	-42	-10+(-42)=-52
-12	-35	-12+(-35)=-47
-14	-30	-14+(-30)=-44
-15	-28	-15+(-28)=-43
-20	-21	-20+(-21)=-41

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-14\"$ and $\"-30\"$ add to $\"-44\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-14\"$ and $\"-30\"$ both multiply to $\"420\"$ and add to $\"-44\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-44x\"$ with $\"-14x-30x\"$ . Remember, $\"-14\"$ and $\"-30\"$ add to $\"-44\"$ . So this shows us that $\"-14x-30x=-44x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"20x%5E2%2Bhighlight%28-14x-30x%29%2B21\"$ Replace the second term $\"-44x\"$ with $\"-14x-30x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%2820x%5E2-14x%29%2B%28-30x%2B21%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"2x%2810x-7%29%2B%28-30x%2B21%29\"$ Factor out the GCF $\"2x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"2x%2810x-7%29-3%2810x-7%29\"$ Factor out $\"3\"$ 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( " $\"%282x-3%29%2810x-7%29\"$ Combine like terms. Or factor out the common term $\"10x-7\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"20x%5E2-44x%2B21\"$ factors to $\"%282x-3%29%2810x-7%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by FOILing $\"%282x-3%29%2810x-7%29\"$ to get $\"20x%5E2-44x%2B21\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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