Parameter Exploration of Bivariate Quadratic Inequality x²+2xy+y²+5x+ay+b>0

This GeoGebra example investigates the solution set of the bivariate quadratic inequality x² + 2xy + y² + 5x + ay + b > 0 and its dependence on parameters a and b. Since x² + 2xy + y² = (x+y)², the inequality can be rewritten as (x+y)² + 5x + ay + b > 0, combining a perfect square with linear terms. Through dynamic geometric construction, it visualizes how the solution region in the coordinate plane changes as parameters a and b vary. The applet incorporates geometric constructions such as rhombi and equilateral triangles alongside tangent line analysis to help students understand the geometric meaning of quadratic inequalities and the constraints imposed by the parameters. The interactive sliders allow learners to adjust a and b in real time, observe the evolving shaded regions, and discover the boundary conditions under which the inequality holds. This makes it a valuable teaching tool for high school or early college courses in analytic geometry and inequality optimization, supporting both conceptual understanding and exploratory learning.