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Good Try--but

Set up problem
X=rate over
X-2 rate back
t is time over
t' is time back
total time is 2.5 (5/2)hrs
total distance is 12 miles and each way is equal

So
Xt=6 & (X-2)t'=6
t+t'=5/2

rearranging slightly

t=6/X & t'=6/(X-2)
t+t'=6/X+6/(X-2)=5/2

findiing common fraction of left side

6/X+6/(X-2)=[6(X-2)+6X]/[X(X-2)]=[6X-12+6X]/[X^2-2X]=[12X-12]/[X^2-2X]

so
[12X-12]/[X^2-2X]=5/2
rearranging some more
2[12X-12]=5[X^2-2X]
24X-24=5X^2-10X
setting =0
5X^2-34X+24=0

this is quadratic eqn
solving quadratic you get 2 possible solutions for X
hint use [-b+/-sqrt(b^2-4ac)]/2a
and you get X=6 & X=0.8

put in conditions
if X=6 trip over is 6mph and trip back is 4mph
if X=0.8 trip over is 0.8mph and trip back is -1.2mph
one of these X's is a meaningless answer so X=6

check answer
trip over is 6/6=1 hr
trip back is 6/4=1.5 hr
& 1+1.5=2.5

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