An equation of the form ax² + bx + c = 0 will have reciprocal roots provided a = c ? 0. If you insist that the roots be real, then you'll need to add the condition b² > 4a². You'll actually find that

b = - a(p² + 1)/p

where the roots of the equation are p and 1/p.

You can drop the condition b² > 4a² and have complex conjugate roots that are reciprocals. Such roots will necessarily have modulus 1.

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