Solving an equation with the quadratic formula involves multiple calculations, some of which may be prone to error. This activity acquaints students with some of the commonly made mistakes and encourages them to write and evaluate expressions carefully. Students critique the reasoning of the presented work and construct arguments for correcting the errors (MP3).

As students work, identify those who can clearly and correctly explain the errors in the worked solutions. Ask them to share their responses during the class discussion.

Display the worked solutions in the Task Statement for all to see. Select previously identified students to identify and explain the error(s) in each worked solution.

After each student presents, ask the class to classify the error(s) by type and to explain their classification. For instance, are the errors careless mistakes or computational mistakes? Do they show gaps in understanding, incomplete communication, or a lack of precision?

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Conclude the discussion by inviting 2–3 students to share a norm they identified in action. Provide this sentence frame to help students organize their thoughts in a clear, precise way:

• “I noticed our norm ‘’ in action today, and it really helped me/my group because .”

This activity prompts students to check the solutions to quadratic equations and to recognize that there is more than one way to do so. Different approaches include substituting solutions to see if the equation is true, solving using a different algebraic method, or solving by graphing.

Monitor for students who use these different strategies to verify the solutions to the equation  . Here are some strategies they may use, from less efficient to more efficient.

• Solve the equation using another algebraic method, and check if they get the same solutions. For instance, if they used the quadratic formula initially, they might choose to solve it again by completing the square or by rewriting the equation in factored form.
• Graph both sides of the equation to find the value of the intersection, or subtract 47 from each side, graph , and check the horizontal intercepts of the graph.
• Substitute the solutions for in the equation and check if the equation is true.

The algebraic strategies are more precise but could lead to errors if the computations are performed incorrectly. The graphing strategies can be accomplished quickly with the use of technology but may give only approximate solutions. 

Making calculators and graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5).

Invite previously selected groups to share their strategies for checking solutions. Sequence the discussion of the strategies in the order listed in the Activity Narrative. If possible, record and display their work for all to see.

Connect the different responses to the learning goals by asking questions such as:

• “What are the advantages and disadvantages of checking by graphing?” (Technology is quick and straightforward to use, but the solutions from technology may be only approximate.)
• “What are the advantages and disadvantages of checking by substituting in the answer?” (It is the most accurate because the solutions are not rounded, but can be difficult to work with and may cause errors when working out the algebra.)
• “If a solution is not rational, like , what are some ways to check by substitution?” (Depending on the calculator technology used, we might be able to input the 2 solutions exactly. If not, we could find a decimal approximation and substitute in the approximate value to check that each side of the equation is very close.)

If no one mentions solving with a different strategy or graphing as a way to verify solutions, bring these up. Display a graph such as this one to show that the graph can immediately show that 74.50 is not a solution to the equation , but 64.50 is very, very close.