Chapter 28: Connecting Factoring with The Quadratic Formula: Situation 22 From the MACMTL–CPTM Situations Project

A teacher who had completed a unit on factoring quadratic polynomials began a unit on the quadratic formula. One student asked whether there was a direct connection between factoring quadratic polynomials and the quadratic formula. The teacher wondered about the different ways one might answer the student's question.

The topic of solving quadratic equations typically involves three processes: (a) factoring, (b) completing the square, and (c) using the quadratic formula. Although each approach can be used to solve any quadratic equation with real coeficients, sometimes one of the approaches is more efficient than the others. For example, some quadratic equations cannot be solved by factoring expressions over the integers but can be solved by factoring over the real numbers. These equations are more easily solved using the quadratic formula. In addition, not every quadratic can be solved by factoring over the real numbers (e.g., x² + 1 = 0). In the three Foci, the benefits of each process are presented as well as the connections among them. In addition, a geometric model is offered to support each algebraic explanation.