examday

/TMUA

Quadratic Chase is a single-player game. To start the game, the player chooses two distinct nonzero integers $a$ and $b$ and writes the equation $x^2+ax+b=0$ on a blackboard.

On each turn, if the equation currently written on the blackboard has two distinct nonzero integer solutions $x=u$ and $x=v$, then the player erases the equation and replaces it with one of the equations $x^2+ux+v=0$ and $x^2+vx+u=0$, of their choosing. Otherwise, the game ends.

There is exactly one initial choice of $(a,b)$ for which the game can be played forever. For this pair, find the value of $a+b$.