Is solving a quadratic equation using Turing machine impossible?
The problem is not clear enough. Turing Machines can or cannot solve problems if those problems are languages, that means, sets of strings over a finite alphabet. You cannot represent all reals in this way, so it is not even a problem that a TM could even attempt to solve. In the same fashion that world hunger is not a problem in the precise sense that a computational complexity requires it to be.

The problem is obviously solvable if you define it as the sets of words that encode three rational numbers a, b and c in a standard way.  How do you solve this? Well, the construction of the machine that does this is rather complicated, but you know how to do this in let's say, Python, right? So you can always compile your Python code into a TM.