SOLUTION: 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 each need to assemble 8

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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
each need to assemble 8 more chairs than usual. How many workers are on the manufacturing crew?
Answer by richwmiller(17219) About Me  (Show Source):
You can put this solution on YOUR website!
x*c=120
x=120/c
(120/c-4)*(c+8)=120
(120/c*c-4*c-32+120/c*8=120
120-4*c-32+960/c=120
-4*c+960/c=32
-4*c^2+960=32c
4c^2+32c-960=0
c=12
x=10
(10-4)*(20)=120
6*20=120
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:



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).