document.write( "Question 910575: A manufacturing crew needs to assemble 120 chairs per day, divided equally among all workers. One day, 4 workers call out sick, and the remaining workers
\n" ); document.write( " each need to assemble 8 more chairs than usual. How many workers are on the manufacturing crew? \n" ); document.write( "
Algebra.Com's Answer #552552 by richwmiller(17219)\"\" \"About 
You can put this solution on YOUR website!
x*c=120
\n" ); document.write( "x=120/c
\n" ); document.write( "(120/c-4)*(c+8)=120
\n" ); document.write( "(120/c*c-4*c-32+120/c*8=120
\n" ); document.write( "120-4*c-32+960/c=120
\n" ); document.write( "-4*c+960/c=32
\n" ); document.write( "-4*c^2+960=32c
\n" ); document.write( "4c^2+32c-960=0
\n" ); document.write( "c=12
\n" ); document.write( "x=10
\n" ); document.write( "(10-4)*(20)=120
\n" ); document.write( "6*20=120
\n" ); document.write( "\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "
Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


\"4%2Ac%5E2%2B32%2Ac-960\" Start with the given expression.



\"4%28c%5E2%2B8c-240%29\" Factor out the GCF \"4\".



Now let's try to factor the inner expression \"c%5E2%2B8c-240\"



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Looking at the expression \"c%5E2%2B8c-240\", we can see that the first coefficient is \"1\", the second coefficient is \"8\", and the last term is \"-240\".



Now multiply the first coefficient \"1\" by the last term \"-240\" to get \"%281%29%28-240%29=-240\".



Now the question is: what two whole numbers multiply to \"-240\" (the previous product) and add to the second coefficient \"8\"?



To find these two numbers, we need to list all of the factors of \"-240\" (the previous product).



Factors of \"-240\":

1,2,3,4,5,6,8,10,12,15,16,20,24,30,40,48,60,80,120,240

-1,-2,-3,-4,-5,-6,-8,-10,-12,-15,-16,-20,-24,-30,-40,-48,-60,-80,-120,-240



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to \"-240\".

1*(-240) = -240
2*(-120) = -240
3*(-80) = -240
4*(-60) = -240
5*(-48) = -240
6*(-40) = -240
8*(-30) = -240
10*(-24) = -240
12*(-20) = -240
15*(-16) = -240
(-1)*(240) = -240
(-2)*(120) = -240
(-3)*(80) = -240
(-4)*(60) = -240
(-5)*(48) = -240
(-6)*(40) = -240
(-8)*(30) = -240
(-10)*(24) = -240
(-12)*(20) = -240
(-15)*(16) = -240


Now let's add up each pair of factors to see if one pair adds to the middle coefficient \"8\":



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First NumberSecond NumberSum
1-2401+(-240)=-239
2-1202+(-120)=-118
3-803+(-80)=-77
4-604+(-60)=-56
5-485+(-48)=-43
6-406+(-40)=-34
8-308+(-30)=-22
10-2410+(-24)=-14
12-2012+(-20)=-8
15-1615+(-16)=-1
-1240-1+240=239
-2120-2+120=118
-380-3+80=77
-460-4+60=56
-548-5+48=43
-640-6+40=34
-830-8+30=22
-1024-10+24=14
-1220-12+20=8
-1516-15+16=1




From the table, we can see that the two numbers \"-12\" and \"20\" add to \"8\" (the middle coefficient).



So the two numbers \"-12\" and \"20\" both multiply to \"-240\" and add to \"8\"



Now replace the middle term \"8c\" with \"-12c%2B20c\". Remember, \"-12\" and \"20\" add to \"8\". So this shows us that \"-12c%2B20c=8c\".



\"c%5E2%2Bhighlight%28-12c%2B20c%29-240\" Replace the second term \"8c\" with \"-12c%2B20c\".



\"%28c%5E2-12c%29%2B%2820c-240%29\" Group the terms into two pairs.



\"c%28c-12%29%2B%2820c-240%29\" Factor out the GCF \"c\" from the first group.



\"c%28c-12%29%2B20%28c-12%29\" Factor out \"20\" 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.



\"%28c%2B20%29%28c-12%29\" Combine like terms. Or factor out the common term \"c-12\"



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So \"4%28c%5E2%2B8c-240%29\" then factors further to \"4%28c%2B20%29%28c-12%29\"



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Answer:



So \"4%2Ac%5E2%2B32%2Ac-960\" completely factors to \"4%28c%2B20%29%28c-12%29\".



In other words, \"4%2Ac%5E2%2B32%2Ac-960=4%28c%2B20%29%28c-12%29\".



Note: you can check the answer by expanding \"4%28c%2B20%29%28c-12%29\" to get \"4%2Ac%5E2%2B32%2Ac-960\" or by graphing the original expression and the answer (the two graphs should be identical).


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