irreducible quadratic trinomial over finite field

Question

In p130 of 《finite field 》 by Lidl et al. : For a trinomial x^2 +x +a over a finite field F_q of odd characteristic it is easily seen it is irreducible over F_q if and only if a is not of the form a = 4^{-1} - b^2, b \in F_q. How can I deduce it and how can I understand 4^{-1}, for example F_q = F_{3^n} with any positive integer n. Does the discriminant over a real field validates over the finite field?