Let the time for Tap A to fill the pool = A hours
Let the time for Tap B to fill the pool = A + 6 hours
So Tap A fills 1/A of the pool in every hour
And Tap B fills 1/ {A + 6) of the pool every hour
Add these and we get that
1 / A + 1 / ( A + 6) = [ A + 6 + A ] / [ A (A + 6)] = [ 2A + 6 ] / [ A^2 + 6A ]
The reciprocal of this is the time it takes both taps working together to fill the pool =
[ A^2 + 6A ] / [2A + 6 ] [ in hours ]
And we know that
Time for both taps working together + 2 hours = = time for Tap A to fill the pool
So we have that
[ A^2 + 6A ] / [2A + 6 } + 2 = A
Simplify
A^2 + 6A / [ 2A + 6 ] = A - 2
A^2 + 6A = [ A - 2 ] [ 2A + 6]
A^2 + 6A = 2A^2 - 4A + 6A - 12
A^2 + - 4A - 12 = 0 factor
(A - 6) ( A + 2) = 0
Setting the first factor to 0 and solving for A gives us the positive answer for A = 6 hours to fill pool
And B takes 6 + 6 = 12 hours to fill the pool
So....the time for both to fill the pool is
1/6 + 1/12
[12 + 6 ] / [ 6 * 12 ]
18/ 72 take the reciprocal
72 / 18 = 4 hours for both to fill the pool