In this activity, students find at least one solution of by substituting different values of , evaluating the expression, and checking if it has a value of zero. Experiencing this inefficient method puts students in a better position to appreciate why it may be desirable to write in factored form and use the zero product property.

If a student found that -5 makes the expression equal 0, ask them to demonstrate that equals 0.

Discuss with students:

• "Can be written in factored form? What are the factors?" (Yes,  and .)
• "If can be written as , can we solve instead?" (Yes.) "Will the solutions change if we use this equation?" (No. The equations are equivalent, so they have the same solutions.)
• "Why might someone choose to rewrite and solve instead?" (Because the expression is equal to 0, rewriting it in factored form allows us to use the zero product property to find both solutions. It may be more efficient than substituting and evaluating many values for . It also makes it possible to see how many solutions there are, which is not always easy to tell when the quadratic expression is in standard form.)

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.

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).