When solving , some students may confuse with and conclude that the solutions are and . Clarify that means 2 times , and that the only thing being squared is the . If both the 2 and are squared, a pair of parentheses is used to group the 2 and the so that we know both are being squared.

In this activity, students encounter quadratic equations that are slightly more elaborate than those in the Warm-up but that can still be solved by reasoning in various ways.

Monitor for students who use these strategies from less efficient to more efficient.

• Substitute different values for , with or without technology.
• Graph each side of the equation and find the intersection.
• Make use of the structure in the equations to solve by algebraic moves or by reasoning about what a squared factor must equal.

Identify students who use these strategies, and ask them to share during the classroom discussion. Alternatively, consider arranging for students who use the same strategy to discuss and then prepare to share their approach.

Students who use technology to solve the equations engage in choosing tools strategically (MP5). Those who solve by analyzing and taking advantage of the composition of the equations practice making use of structure (MP7).

Invite previously selected groups to share their strategies for solving the equations. 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:

• “How is graphing similar to substituting values into the equations?” (When we graphed, the computer substituted a lot of values quickly and represented them visually as a curve or line.)
• “When we do an algebraic move like adding 30 to each side of the equation, how would that affect the graph of each side? Why does it not change the solution when graphing?” (It would shift both graphs up 30 units. Because both graphs are shifted vertically, their intersection has the same -coordinate.)

If no students reasoned about the solutions algebraically, be sure to demonstrate it (without saying “take the square root of each side”) and to record the reasoning process for all to see. For example:

• We can interpret as “something plus 4 is 404.” That “something” must be 400, so we can write . This equation means “something times itself is 400.” That “something” must be 20 or -20, because they each give 400 when squared. The two solutions are therefore 20 and -20.

  Point out that the reasoning that took us from to gave the same equation as subtracting 4 from each side of the original equation.

• We can see as “144 is something squared,” so the “something” is either 12 or -12. We can represent this with and . The solutions are 11 and -13.