Completing the Square (Part 1): Visual Decomposition into a Perfect Square

This example demonstrates the key idea of completing the square using an interactive geometric decomposition: it visually transforms a quadratic expression into an equivalent perfect-square form by matching the relationships between segment lengths (or areas). The mathematical concepts include completing the square for quadratic expressions, the perfect-square identity, and the algebraic equivalence of ax^2+bx+c with a(x−h)^2+k, linked to geometric interpretation. Construction highlights from the GeoGebra commands and objects: Polygon is used to draw the subdivided polygon regions that represent the quadratic pieces and the “added/removed” square components; Sequence generates or updates critical points/stepwise parameter changes; Textfield provides dynamic labels for coefficients and intermediate values; and interactive controls (buttons) tied to parameters (e.g., a, b, c and derived quantities) together with coordinate/axis-aligned elements (hori/vert) support the step-by-step visual completion process. Suitable for middle and high school algebra instruction and for teachers or students exploring why the vertex form of a quadratic comes from completing the square in a dynamic geometry setting.