For p, q ∈ R, consider the real valued function f(x)=(x- p )^2- q, x ∈ R and q >0. Let a_1, a_2, a_3 and a_4 be in an arithmetic progression with…

For $\displaystyle \mathrm{p}, \mathrm{q} \in \mathbf{R}$, consider the real valued function $\displaystyle f(x)=(x-\mathrm{p})^2-\mathrm{q}, x \in \mathbf{R}$ and $\displaystyle \mathrm{q}>0$. Let $\displaystyle \mathrm{a}_1, \mathrm{a}_2$, $\displaystyle \mathrm{a}_3$ and $\displaystyle \mathrm{a}_4$ be in an arithmetic progression with mean p and positive common difference. If $\displaystyle \left|f\left(\mathrm{a}_i\right)\right|=500$ for all $\displaystyle i=1,2,3,4$, then the absolute difference between the roots of $\displaystyle f(x)=0$ is \_\_\_\_.