In this section, students are presented with activities that motivate the need for more algebraic methods for solving quadratic equations. Students write quadratic equations from situations, but recognize that they do not have the tools to accurately solve the equations unless they are in very specific forms, such as a factored form equal to zero or a perfect square equal to another perfect square.

In this section, students learn to complete the square as an additional strategy for solving quadratic equations. First, they begin by recognizing when quadratic expressions can be rewritten as perfect squares and how this form is useful for solving equations of the form . Then, students learn how to turn any quadratic equation into that form by completing the square.

The final section of this unit is a lesson where students have the opportunity to apply their thinking from throughout the unit. As this is a short section followed by an End-of-Unit Assessment, there are no section goals or checkpoint questions.

In this section, students look more deeply into solving quadratic equations that can be written as a factored form equal to zero. They formalize their understanding of the zero product property, then work on recognizing patterns that allow them to rewrite quadratic expressions from standard form to factored form. Students also begin to recognize that quadratic equations can have 0, 1, or 2 solutions.

Note that this course includes only a few lessons on transforming quadratic expressions from standard form to factored form. This is intentional. The goal is to help students see factored form conceptually, understand what it can tell us about the function, and use that knowledge in modeling problems, including problems where the zeros are not rational and algebraic factoring wouldn't help. Too much attention to the algebraic skill of factoring can obscure these underlying concepts.

Sometimes—including in later courses and beyond—expressions do need to be written in factored form. With the increasing availability of computer algebra systems, however, there is less need to spend lots of time learning how to factor by hand.

In this section, students encounter the quadratic formula. Initially, the formula is given to students, and they notice that the solutions match those found using other methods. After getting comfortable with using the formula, they examine a proof of the quadratic formula that uses completing the square.

An optional lesson in which students analyze common errors encountered using the quadratic formula is included. If students are also making similar mistakes, consider making time to include the lesson.

The section ends with a look at rational and irrational numbers first encountered in grade 8. Here, they classify sums and products of various combinations of these numbers as either irrational or rational.

In this section, students revisit the vertex form of quadratic expressions. In a previous unit, students noticed that this form can be used to easily find the vertex of a quadratic function and identified that point on a graph. Here, students use the process of completing the square to rewrite expressions from standard form to vertex form, then classify the vertex as either a maximum or minimum and justify their reasoning.