The most commonly used tool is modular arithmetic. Another common treatment of impossibility is to place a part of the Diophantine equation (let's say on the left hand side of the equation), and then show that it lies between consecutive terms of a sequence described by the right hand side of the equation. This treatment relies heavily on clever algebraic manipulations.

Finally if a question asks for infinitely many solutions, then the solutions can be found by use of an appropriate algebraic identity or by use of successive applications of the quadratic formula. One can also try to find a recursive sequence of solutions as in the case of the Pell's equation.

Warming up example: Show that the equation