This task prompts students to try different ways to solve a quadratic equation. They are familiar with solving equations by performing the same operation on each side of an equation, but here they see that this is not really a workable strategy. Students are also discouraged from using a graph to solve the equation. Because they do not yet know an efficient way to use algebra to solve , they need to try different strategies and persevere in problem solving (MP1).

As students work, notice those who use strategies listed in the Activity Synthesis. Ask them to share their approach during discussion.

Poll the class on their solutions to the equation, and record and display the solutions for all to see. Then, ask some students to share their strategies and any associated challenges. If not mentioned by students, discuss the limitations of these approaches:

• Isolating the variable: If we try to solve by performing the same operation on each side of the equation, we quickly get stuck.

But then what? If we add to each side, we now have a variable on both sides and cannot combine any like terms. We could multiply or divide each side by any constant we wish, but we are no closer to isolating .

• Guessing and checking: We can evaluate the quadratic expression at different values of until the expression has a value that is 0 or close to 0. For example, when is 4, the expression has a value of 128. At , it has a value of 64, and at , it has a value of -32. That means is between 5 and 6, so we need to try different decimal values in that range.

  This process is laborious and may not get us to a precise solution.

• Graphing: Students may suggest that a graph would allow them to solve the problem much more quickly. Use graphing technology to demonstrate that if we graph the equation , an approximate solution given is 5.702, as shown in the image.

  

  If we evaluate , however, we get about -0.044864, rather than exactly 0.

  A graph is useful for approximating values, but it isn’t always possible to use it to find exact values.

Tell students that in this unit they will learn some efficient strategies for solving equations like these.

This activity aims to show that it is relatively easy to solve a quadratic equation when one side of the equation is zero and the other side is a quadratic expression in factored form, and that it may be a little tricky to solve the equation otherwise.

Although students are not yet formally introduced to the zero product property, they do have experience finding the zeros of a quadratic function when given an expression in factored form. This prior knowledge enables them to reason about the solutions to the equations.

As students make sense of the equations and ways to use them to solve a contextual problem, they practice reasoning quantitatively and abstractly (MP2).

If students struggle to connect the expressions that define the function to the questions, ask them what the input and output of the function represent. If students struggle with the first question, ask them what values of would yield a value of 0 for the expression.

Invite students to share their responses and strategies. Make sure students see that the first question can be represented by solving the equation , and that the second question can be represented by solving either or .

Ask students,

• “How can you show, without graphing, that will produce no revenue, or that it is a solution to the equation ?” (When , the factor is 0, and therefore the entire expression equals 0.)
• “How can you show that will also produce no revenue, or that it is also a solution to the same equation?” (When , the factor is 0, and likewise, the entire expression is 0.)
• “Can we use the same reasoning to find the solutions to or ? Why or why not?” (No, not easily. Neither equation shows the zeros of a function. There are many pairs of factors that have 500 as a product.)

Make sure students see that it is fairly straightforward to find the solutions to equations such as , but the same cannot be said about equations such as  or .

Highlight that all the equations in this activity are quadratic equations. Explain that a quadratic equation is one that can be written in the form of , and where is not 0. 

If time permits, ask students to show how all of the equations seen here can be written in this form.