Two constructors working together finished building a room in 6 days. how long would it take for each constructor to build the room by himself, if it is known that one of them would require 9 more days than the other?

It would take one construction worker 9 days and the other 18 days.

Let x be the number of days the first construction worker takes to build the room by himself. If x were 2, he could build 1/2 of the room in the time limit, etc ... so 1/x will be the portion of the room he can build by himself in the given tie limit.

The second construction worker can build 1 / (x+9) of the room by himself in the given time limit.

Together, we have the equation

1/x (6) + 1 / (x+9) (6) = 1

[The speed of the first contractor times the number of days, and the speed of the second contractor times the number of days; together they build 100% of the room]

This gives us

6/x + 6 / (x+9) = 1

We will multiply everything by x to get it off of the denominator:

6/x (x) + (6 / (x+9)) (x) = x

6 + 6x / (x+9) = x

Multiply everything by x+9 now:

6 (x+9) + (6x / (x+9)) (x+9) = x (x+9)

6x + 54 + 6x = x² + 9x

12x + 54 = x² + 9x

Subtract 12x from each side:

12x + 54 - 12x = x²+9x-12x

54 = x²-3x

Subtract 54 from each side:

54-54 = x²-3x-54

0 = x²-3x-54

This factors easily; - 9 (6) = - 54 and - 9+6 = - 3:

0 = (x-9) (x+6)

Using the zero product property we know either x-9=0 or x+6=0; this gives us x=9 or x=-6. Negative time makes no senses, so x=9 hours.

This means the slower contractor takes 9+9 = 18 hours.