I am learning about a Bell State, and am trying to show that they are entangled. I believe that the required proof is to show that the system
$$\alpha_0^2+\alpha_1^2=1$$ $$\beta_0^2+\beta_1^2=1$$ $$\alpha_0\beta_0=1/\sqrt{2}$$ $$\alpha_1\beta_1=1/\sqrt{2}$$
has no solutions. I have tried various ways of rewriting each variable in terms of others etc. without success. Any hints?
EDIT: I apologize, I should have made clear. In QM probabilities can be complex - what exactly this means intuitively is unclear to me, but algebraically it means that $\alpha_i,\beta_i$ can be complex.