One positive integer is 55 less than twice another. The sum of their squares is 185. Find the integers.

Let one integer =  x

Let  the other  =  2x  -  55

So

x^2   +  (2x  - 55)^2    =   185

5x^2  -  220x  +  3025   =185

5x^2  -220x  +   2840  =  0

The  discriminant    is

220^2   - 4 * 5 * 2840    =  -8400

So.......this has  no real solution