In this activity, students solve a variety of quadratic equations by integrating what they learned about rewriting quadratic expressions in factored form and their understanding of the zero product property. They begin by analyzing and explaining the steps in a solution strategy and then applying their observations to solve other equations, both of which require sense making and perseverance (MP1). Students practice attending to precision (MP6) as they study solution steps and communicate what each step does or means.

As students work, look for the equations many students find challenging and those on which errors are commonly made. Discuss these challenges and errors during the Activity Synthesis.

Consider displaying the solutions for all to see and discussing only the equations that students found challenging and any common errors.

The last equation is unlike most equations students have seen. Invite students to share how they solved that equation. Discuss questions such as:

• “Can we use the zero product property to write and ? Why or why not?” (No. The zero product property can be applied only to products that equal 0. The expression has a product for one of its terms, but the expression itself is a difference.)
• “How can it be solved, other than by graphing?” (We can expand the factored expressions using the distributive property and write an equivalent equation: , or . This last equation can then be written as , which allows it to be solved.)

This activity reinforces what students learned earlier about the connections between the solutions of a quadratic equation and the zeros of a quadratic function. Previously, students were given equations and asked to graph them to determine the number of solutions and the values of the solutions. Here, they are prompted to work the other way around: to write an equation to represent a quadratic function with only one solution. To do so, students need to make use of the structure of the factored form and their knowledge of the zero product property (MP7).

Invite students to share how they solved the equation algebraically. Next, invite students to share the equations they generated. Record and display them for all to see.

Students most likely have written equations in the form of . Ask students why the factored form, rather than the standard form, may be preferred. Highlight that by using the same expression for the two factors, we know that the solution to will be a single number.

Ask students to describe the graph of a quadratic function with one solution. Point out that this means that the function will have only one zero, and the graph of the function will have a single horizontal intercept.