JEE Main 2022 · 28 Jul · Shift 1, Mathematics question 26

For $p,q\in\mathbf R$, consider the real valued function $f(x)=(x-p)^2-q$, $x\in\mathbf R$ and $q>0$. Let $a_1$, $a_2$, $a_3$ and $a_4$ be in an arithmetic progression with mean p and positive common difference. If $|f(a_i)|=500$ for all $i=1,2,3,4$, then the absolute difference between the roots of $f(x)=0$ is ______.