The splitting fields of two irreducible polynomials over $Z / p Z$ both of degree 2 are isomorphic $p$ is a prime. Let $ f_1, f_2 \in Z / p Z [t]$ both of degree 2 and irreducible. Show that they have isomorphic splitting fields.

My approach was let $ K_1 = F(\alpha_1, \beta_1) / F$ be the splitting field of $f_1$, and let $K_2 = F( \alpha_2, \beta_2) / F$ be the splitting field of $ f_2$. Then I have trouble finding any relations between $ \alpha$'s and $\beta$'s. Any help is appreciated.

Some ideas (hopefully you've already studied this stuff's details):
For a prime $\;p\;$ and for any $\;n\in\Bbb N\;$, prove that the set of all the roots of the polynomial $\;f(x):=x^{p^n}-x\in\Bbb F_p[x]\;$ in (the, some) algebraic closure $\;\overline{\Bbb F_p}\;$ of $\;\Bbb F_p:=\Bbb Z/p\Bbb Z\;$ , with the usual operations modulo $\;p\;$ , is a field with $\;p^n\;$ elements, which we denote by $\;\Bbb F_{p^n}\;$
From the above it is immediate that $\;\Bbb F_{p^n}\;$ is the minimal field which contains all the roots of $\;f(x)\;$ and is thus this polynomial's splitting field over $\;\Bbb F_p\;$ .
Now, apply the above to the particular case $\;n=2\;$ and deduce at once your claim.
