Peter picked three non zero ***real*** numbers $a,b,c$ and Alan arranges these numbers as coefficients of Quadratic equation $ax^2+bx+c=0$.

Peter wins if this Quadratic has distinct Rational roots, else Alan wins. 

How can we prove that Peter always has winning strategy.

Since $a,b,c$ are non zero reals we have roots of quadratic as

$$\frac{-b \pm \sqrt{b^2-4ac}}{2a}$$

Now Peter can win game only if $$b^2-4ac=k^2$$ where $k$ is Integer. But even though $b^2-4ac=k^2$ is integer, how can we say roots are rational?