2. Student Xiao Ming randomly selects an arithmetic sequence $\left\{a_{n}\right\}$ with all terms being non-zero integers, and then lists 2020 quadratic equations in one variable $x$: $E_{i}: a_{i} x^{2}+a_{i+1} x+a_{i+2}=0(i=1,2, \cdots, 2020)$, and writes all the roots of each equation on the blackboard. Find the maximum possible number of distinct real roots on the blackboard.
Let's assume the common difference of the arithmetic sequence $\left\{a_{n}\right\}$ is $d$, clearly $d \neq 0$, otherwise $x^{2}+x+1=0$ has no solution.
Thus, $\left(a_{i+1}-d\right) x^{2}+a_{i+1} x+\left(a_{i+1}+d\right)=0$
$$
\begin{array}{l}
\Rightarrow \Delta=a_{i+1}^{2}-4\left(a_{i+1}-d\right)\left(a_{i+1}-d\right) \Rightarrow-3 a_{i+1}^{2}+4 d^{2}>0 \\
\Rightarrow a_{i+1}^{2}<\frac{4}{3} d^{2} \Rightarrow a_{i+1} \in\left[-\frac{2}{\sqrt{3}}|d|, \frac{2}{\sqrt{3}}|d|\right] .
\end{array}
$$

Notice that the length of this interval is $2|d|<\frac{4|d|}{\sqrt{3}}<3|d|$, and the distance between adjacent terms of $\left\{a_{n}\right\}$ is $|d|$, so there are at most three terms of $\left\{a_{n}\right\}$ in the interval $\left[-\frac{2}{\sqrt{3}}|d|, \frac{2}{\sqrt{3}}|d|\right]$.
The roots of any two different equations must be different, otherwise, subtracting these two equations and eliminating multiples of $d$ would result in $x^{2}+x+1=0$ having no solution, which is a contradiction. Therefore, the maximum possible number of distinct real roots on the blackboard is 6.

On the other hand, let $d=7$, then the interval is $(-8.08,8.08)$, taking $a_{2}, a_{3}, a_{4}=-8,-1,6$, the sequence is $a_{n}=7 n-22\left(n \in \mathbf{N}^{*}\right)$, the corresponding three equations are
$$
-15 x^{2}-8 x-1=0,-8 x^{2}-x+6=0,-x^{2}+6 x+13=0 \text {. }
$$

Their roots are $-\frac{1}{5},-\frac{1}{3}, \frac{-1 \pm \sqrt{193}}{16}, 3 \pm \sqrt{22}$, a total of 6.