document.write( "Question 251553: How would you factor 5x^2-42x+16 using the box method? \n" ); document.write( "

Algebra.Com's Answer #183238 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
I'm not sure what you mean by the box method, but here's one way to do it.\r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"5x%5E2-42x%2B16\"$ , we can see that the first coefficient is $\"5\"$ , the second coefficient is $\"-42\"$ , and the last term is $\"16\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"5\"$ by the last term $\"16\"$ to get $\"%285%29%2816%29=80\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"80\"$ (the previous product) and add to the second coefficient $\"-42\"$ ?\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 $\"80\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"80\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,4,5,8,10,16,20,40,80\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-4,-5,-8,-10,-16,-20,-40,-80\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 $\"80\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*80 = 80
\n" ); document.write( "2*40 = 80
\n" ); document.write( "4*20 = 80
\n" ); document.write( "5*16 = 80
\n" ); document.write( "8*10 = 80
\n" ); document.write( "(-1)*(-80) = 80
\n" ); document.write( "(-2)*(-40) = 80
\n" ); document.write( "(-4)*(-20) = 80
\n" ); document.write( "(-5)*(-16) = 80
\n" ); document.write( "(-8)*(-10) = 80\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 $\"-42\"$ :\r
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First Number	Second Number	Sum
1	80	1+80=81
2	40	2+40=42
4	20	4+20=24
5	16	5+16=21
8	10	8+10=18
-1	-80	-1+(-80)=-81
-2	-40	-2+(-40)=-42
-4	-20	-4+(-20)=-24
-5	-16	-5+(-16)=-21
-8	-10	-8+(-10)=-18

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-2\"$ and $\"-40\"$ add to $\"-42\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-2\"$ and $\"-40\"$ both multiply to $\"80\"$ and add to $\"-42\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-42x\"$ with $\"-2x-40x\"$ . Remember, $\"-2\"$ and $\"-40\"$ add to $\"-42\"$ . So this shows us that $\"-2x-40x=-42x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"5x%5E2%2Bhighlight%28-2x-40x%29%2B16\"$ Replace the second term $\"-42x\"$ with $\"-2x-40x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%285x%5E2-2x%29%2B%28-40x%2B16%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%285x-2%29%2B%28-40x%2B16%29\"$ Factor out the GCF $\"x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%285x-2%29-8%285x-2%29\"$ 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( " $\"%28x-8%29%285x-2%29\"$ Combine like terms. Or factor out the common term $\"5x-2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"5x%5E2-42x%2B16\"$ factors to $\"%28x-8%29%285x-2%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"5x%5E2-42x%2B16=%28x-8%29%285x-2%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%28x-8%29%285x-2%29\"$ to get $\"5x%5E2-42x%2B16\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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