What's special about seven?

Algebra Level 5

Consider the following polynomial equation:\[(x^2+a)^2+a=x\]Where \(a\) is a real number such that the equation has four distinct real roots. If the difference between the largest and the smallest roots of equation is \(7\), then \(a\) can be written in the simplest form as:\[-\frac{\overline{ABBB}}{\overline{ACC}}\]Find the value of \(A^2+B^2+C^2\).

Details and Assumptions:

  • \(A\), \(B\) and \(C\) are some digits from \(1\) to \(9\).

  • As an explicit example, let's say that \(-2\), \(-1\), \(0\) and \(1\) are the roots of polynomial, then the largest root is \(1\) and the smallest is \(-2\) and the difference between them is \(1-(-2)=3\).

This problem is original and belongs to this set.

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