This paper aims to provide precise formula solutions of the two-sided quadratic quaternionic equation [Formula: see text]. The main method is to introduce an efficient way of eliminating the number of real parameters from [Formula: see text]. By using the Lefschetz fixed point theorem, we can show that any binomial polynomial quaternionic equation has a non-zero solution, and this result cannot be further generalized.
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Liu et al. (2026) studied this question.
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