2026, Vol. 11, Issue 1, Part B

Algebra and geometry are often taught as distinct branches of mathematics, leading students to perceive symbolic manipulation and visual reasoning as unrelated skills. This study presents an innovative geometric approach to solving quadratic equations of the form ax² + bx + c = 0 (a ≠ 0) using only a compass, straightedge, and Cartesian plane construction. The background of the study is rooted in the need to enhance conceptual understanding and student engagement in secondary-level algebra by bridging abstract algebraic formulas with intuitive geometric representations.
The primary objective of the study is to demonstrate a generalized construction method that visualizes the roots of any quadratic equation through circle geometry, thereby linking the coefficients of the equation to spatial coordinates. Methodologically, the equation is first normalized into monic form, and two fixed points A at (0, 1) and B at (−b/a, c/a) are plotted on the coordinate plane. A circle is then constructed with diameter AB, and the intersections of this circle with the x-axis represent the real solutions of the quadratic equation. The results show that this method consistently yields correct roots across a wide range of cases, including equations with positive, negative, fractional, repeated, and complex roots. The nature of the discriminant is visually interpreted through the circle’s interaction with the x-axis, providing immediate geometric insight into the existence and type of roots. Classroom implementation further indicates improved student motivation, conceptual clarity, and confidence.
The study concludes that the compass-based geometric construction transforms quadratic equations from a procedural task into a meaningful visual exploration. By integrating algebraic reasoning with geometric intuition, this approach supports deeper mathematical understanding and promotes the appreciation of mathematics as both a logical and artistic discipline.