Find the exact value of x so that the equation has exactly 2 solutions when solved for a.
You mean, I take it, 2 distinct real solutions. In that case, the equation must be of the form $\displaystyle (a-u)^2(a-v)= a^3- a- x= 0$. for some u and v.
Multiplying that out, we get $\displaystyle a^3- (2u+v)a^2+ (u^2+ 2uv)a- u^2v= a^3- a- x= 0$. We must have 2u+v= 0, $\displaystyle u^2+ 2uv= -1$ and $\displaystyle u^2v= x$. Solve the first two equations for u and v and then get x from the third equation.