If  $\alpha ,\beta ,\gamma$ are distinct integers, then minimum value of $\left|\Delta \right|$  is equal to, 
Where   $\Delta =\left|\begin{array}{ccc}3{\alpha }^2& {\alpha }^2+\alpha \beta +{\beta }^2& {\alpha }^2+\alpha \gamma +{\gamma }^2\\ {\beta }^2+\alpha \beta +{\alpha }^2& 3{\beta }^2& {\beta }^2+\beta \gamma +{\gamma }^2\\ {\gamma }^2+\alpha \gamma +{\alpha }^2& {\gamma }^2+\beta \gamma +{\beta }^2& 3{\gamma }^2\end{array}\right|$

Splitting into two determinants
$$\begin{gathered} \left|\begin{array}{ccc}{\alpha }^2& \alpha & 1\\ {\beta }^2& \beta & 1\\ {\gamma }^2& \gamma & 1\end{array}\right|\times \left|\begin{array}{ccc}1& 1& 1\\ \alpha & \beta & \gamma \\ {\alpha }^2& {\beta }^2& {\gamma }^2\end{array}\right|=\left|\begin{array}{ccc}3{\alpha }^2& {\alpha }^2+\alpha \beta +{\beta }^2& {\alpha }^2+\alpha \gamma +{\gamma }^2\\ {\beta }^2+\alpha \beta +{\alpha }^2& 3{\beta }^2& {\beta }^2+\beta \gamma +{\gamma }^2\\ {\gamma }^2+\alpha \gamma +{\alpha }^2& {\gamma }^2+\beta \gamma +{\beta }^2& 3{\gamma }^2\end{array}\right| \\ -(\alpha -\beta )(\beta -\gamma )(\gamma -\alpha )(\alpha -\beta )(\beta -\gamma )(\gamma -\alpha ) \end{gathered}$$

$\Delta =-{(\alpha -\beta )}^2{(\beta -\gamma )}^2{(\gamma -\alpha )}^2$$\Rightarrow$   maximum value of $\Delta$  is  $-4$