A geometric method for solving quadratic equations

The theory of quadratic equations (with real coefficients) is an important topic in the secondary school mathematics curriculum, including the current DSE curriculum in Hong Kong. Usually students are taught to solve a quadratic equation ax² + bx + c = 0 ( a ≠ 0 ) algebraically (by factorisation, completing the square and using the quadratic formula), graphically (by plotting the graph of the quadratic polynomial y = ax² + bx + c to find the x-intercepts, if any), and numerically (by the bisection method or Newton-Raphson method). Less well-known is that we can indeed solve a quadratic equation geometrically by geometric construction tools. In this workshop we are going to describe this approach. However, the tool we use is a set square rather than a ruler and a compass, which are the traditional tools employed in geometric construction. We choose a set square because it is more convenient (one tool is used instead of two). Surely the methods to be presented in the workshop can also be carried out in the traditional manner. It is also worth mentioning that any construction done with a ruler and a compass can also be done with a set square, though the underlying geometric theorems may be different. Of course, we cannot draw a circle with a set square. But we can locate as many points as we wish on the circumference of a circle with given centre and radius. We hope that school teachers can incorporate some of the ideas in the design of learning activities for their students.