Solve quadratic equations with complex roots
Apply knowledge of complex numbers to solve quadratic equations with complex solutions (N-CN.C.7).
In this final section of the unit, students connect their knowledge of complex numbers to their previous work with quadratics. They learn that quadratic equations always have complex solutions even when they have no real solutions, unifying the class of quadratic equations when the complex number system is considered.
WHAT: The task presents the work of a student who is solving a quadratic equation by completing the square and ends up faced with a negative number on the right hand side. It then asks the students to solve two equations with non-real solutions by completing the square.
WHY: The goal of this task is to show that extending the number system opens up new possibilities for old methods, in this case the method of completing the square. Students can also use the quadratic formula to solve these equations and see directly how the availability of square roots of negative numbers changes the way they use that formula. The teacher may wish to have students graph the solutions so that they can see that the imaginary solutions are reflections of one another about the real axis.
This activity provides an example where complex numbers arise in an area of mathematics that students previously might not have expected. It is a good example of connections between seemingly different topics in mathematics. This is also an opportunity for students to attend to precision when discussing solutions to a quadratic now that they have the language to do so with MP.6.