Quadratic Formula and Complex Solutions Visualization

This example demonstrates how the quadratic formula is used to obtain the roots of a one-variable quadratic equation and how the situation changes when the discriminant is negative, leading to complex (non-real) solutions. It focuses on the key ideas of the discriminant Δ = b^2 − 4ac, the root expression x = (−b ± √Δ)/(2a), and the interpretation of √(negative number) in terms of the imaginary unit i to produce complex solutions. Construction-wise, the typical GeoGebra workflow is to enter the quadratic equation/parameters to compute and display Δ and the corresponding root formulas, while simultaneously using the quadratic function graph to check real root existence via x-intercepts (not crossing the x-axis when Δ < 0). The provided GeoGebra “construction summary” is empty, so the description emphasizes the standard approach: algebraic computation of roots combined with graphical evidence from the parabola. Suitable for late secondary school and introductory college algebra, especially for lessons on quadratic equations, discriminant-based reasoning, and the motivation for complex numbers.