Abdul's Interchange Theory: A Classroom Formulation of Reciprocal Roots in Quadratic Equations

This paper presents a classroom formulation of the effect of interchanging the leading coefficient a and the constant term c in a quadratic equation ax^2 + bx + c = 0, while keeping b fixed. The formulation, here named Abdul's Interchange Theory, shows that the discriminant is preserved, that both roots are mapped to their reciprocals, and that a root appears unchanged if and only if it is 1 or -1. The paper gives elementary proofs suitable for secondary-school factorisation, a precise domain of validity (a != 0, c != 0), and five worked examples. The author independently observed the pattern through factorisation; the underlying reciprocal-polynomial fact is classical and is acknowledged in Section 2.