What's special about seven?

Algebra Level 5

Consider the following polynomial equation:(x2+a)2+a=x(x^2+a)^2+a=xWhere aa 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 77, then aa can be written in the simplest form as:ABBBACC-\frac{\overline{ABBB}}{\overline{ACC}}Find the value of A2+B2+C2A^2+B^2+C^2.

Details and Assumptions:

  • AA, BB and CC are some digits from 11 to 99.

  • As an explicit example, let's say that 2-2, 1-1, 00 and 11 are the roots of polynomial, then the largest root is 11 and the smallest is 2-2 and the difference between them is 1(2)=31-(-2)=3.

This problem is original and belongs to this set.

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