Quadratic Equations: Geometry of Coefficient–Solution Relationships

① This interactive GeoGebra example visualizes the geometric relationships between the coefficients (a, b, c) of a quadratic equation ax²+bx+c=0 and its solutions (roots), accompanied by algebraic exploration. ② Core mathematical concepts include quadratic equations, intersections of parabolas and lines, tangent and secant lines, similar triangles, fractions, and rectangle areas, linking symbolic algebra with geometric intuition. ③ Construction highlights: uses GeoGebra commands Point, Polygon, Segment, and Intersect to build points, polygons, segments, and intersection points; sliders (aSli, bSli, cSli) and animation buttons (btnForward, btnBack, btnRewind) control coefficient changes, while text objects (text1, text2, etc.) display instructions and numerical data for interactive exploration. ④ Suitable for middle or high school algebra/geometry classes to understand the relationship between roots and coefficients, and for teacher demonstrations or student self-study, deepening comprehension of completing the square, the discriminant, and Vieta's formulas from a geometric perspective.