Hello all!

Once again, I am stuck on a problem, and no matter how I twist it, I always end up in equations that have no solutions.

Question:

The sum of three consecutive members of an arithmetic progression is 2, and the sum of the squares of these members is .

My solution(done with the help of the book's hints)

Let be the "middle" number of the three numbers.

Then

ans if we substitute , we get

But then the equation has no solutions.

All help is appreciated. Anyway, why is it so that some equations do not end up having solutions?