For p, q ∈ R, consider the real-valued function f(x) = (x - p)^2 -q, x ∈ R and q greater than 0. ...

For p, q ∈ R, consider the real-valued function f(x) = (x - p)^2 -q, x ∈ R and q greater than 0. Let a1 , a2 , a3 , and a4 be in an arithmetic progression with mean p and positive common difference. If |f(a1)| = 500 for all i = 1, 2, 3, 4, then the absolute difference between the roots of f(x) = 0 is
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