Together, Tim and Alan can unload a truck 6 in minutes. When Tim works alone, it takes him 9 minutes longer than Alan to unload the truck. How long does it take each person to unload the truck alone?

So, I simplified the denominator, So I had this:T(T+9)/2T+9=6.
Then, I multiplied the denominator on both sides. T(T+9)=6(2T+9).
I distributed.T(T+9)=12T+54
Then...Now here is the part I am not sure about...I distribute the T: T^2+9T=12T+54.
I moved the 9T to the right side: T^2=12T-9T+54.
So now I have this: T^2=3T+54
I am not sure what to do next. How would I go about it? thanks!!!!

Question

Together, Tim and Alan can unload a truck 6 in minutes.   When Tim works alone, it takes him 9 minutes longer than Alan to unload the truck.   How long does it take each person to unload the truck alone?

So, I simplified the denominator, So I had this:T(T+9)/2T+9=6.
Then, I multiplied the denominator on both sides. T(T+9)=6(2T+9).
I distributed.T(T+9)=12T+54
Then...Now here is the part I am not sure about...I distribute the T: T^2+9T=12T+54.
I moved the 9T to the right side: T^2=12T-9T+54.
So now I have this: T^2=3T+54
I am not sure what to do next. How would I go about it? thanks!!!!

dumbsearch2 · Answer

So I found 9 for Tim after solving the quadratic. But I am sort of confused about Alan. If I understand correctly, T=9+A. But if this is the case, then A is 0. Is this even possible? Thanks again for your help. :)

mertsj · Answer

$$\frac{1}{t}+\frac{1}{t+9}=\frac{1}{6}$$
$$6t+54+6t=t^2+9t$$
$$t^2-3t-54=0$$
$$(t-9)(t+6)=0$$
t=9=Alan's time
t+9=18 = Tim's time