According to the meaning of the question, the dyed part is (x-0.78)/2 when it is first inserted into the dye vat.

The dyeing part is [(x-0.78)/2]*2=x-0.78 when the dyeing vat is inserted for the second time.

The undyed part is: x-(x-0.78)=0.78.

Because the dyed part is more than the undyed part at this time1.25m, therefore:

(x-0.78)-0.78= 1.25

Solution:

X = 2.8 1m

So the bamboo pole is 2.8 1 m long.