An Analysis of the Roots of Quadratic Equations Using a Derivative-Based Approach A Comparison with the Discriminant and Completing-the-Square Methods

An Analysis of the Roots of Quadratic EquationsUsing a Derivative-Based Approach AComparison with the Discriminant andCompleting-the-Square Methods In this paper, an analytical and educational method for studying the roots ofquadratic equations is presented based on the derivative of the function, thedivision of the derivative-based reasoning from the original function, and theanalysis of the function’s remainder at its critical point. Unlike the classicaldiscriminant method, which relies solely on an algebraic quantity, the proposedderivative-based approach determines both the type and the numerical values ofthe roots through the behavior of the function itself. To illustrate this method,three numerical examples are examined, corresponding to equal real roots, distinct real roots, and complex roots. In each case, the equation is solved usingthe discriminant method, the derivative-based method, and the completingthe-square method, and the results are compared. The findings show that thederivative-based approach is fully consistent with classical methods while offering a deeper conceptual and educational understanding of quadratic equations.