Can All Coefficients 'a' and 'b' Be Roots in Their Own Set of Quadratics?

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The discussion revolves around whether all coefficients 'a' and 'b' can serve as roots in a set of 'n' distinct quadratics of the form x^2 + a_ix + b_i. It has been established that this is feasible for n=2, but challenges arise for n greater than 3 due to the lack of fixed positions for each value. The participants note that all values of 'a' and 'b' must be distinct, and both real and complex numbers are considered acceptable roots. Clarification is sought regarding any additional constraints that may apply to the roots. The conversation highlights the complexities involved in extending the solution beyond two quadratics.
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1. Homework Statement :
We have 'n' quadratics is the form x^2 +aix+bi
All values of A and b are different.
Is it possible to have all values of A and B as the roots of the n quadratics

2. The attempt at a solution:
Well. I know that all values of 'a' and 'b' must (if possible) be in one of the n (x-t)(x-s) where t and s are two values of a and/or b. I have proven that when n=2 its possible for all values, however i have difficulty with n>3
 
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Could you please clarify the question? Are there any other constraints to the question (like the roots must be real/complex)?

Why should there be a problem when n>3?
 
Well there are no other restraints. Only all the values of a and b are different. I assume that both complex and real numbers are possible.

The answer to your second question I was having difficulty with N>3 is because there are not fixed 'positions' for each value
 

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