In this Warm-up, students begin to think more abstractly about the process of solving quadratic equations. They recognize that some quadratic equations have one solution and others have two.

All of the equations can be solved by reasoning and do not require formal knowledge of algebraic methods, such as rewriting into factored form or completing the square. For example, for , students can reason that must be 4 because that is the only number that gives 3 when subtracted by 1.

Finding the solutions of these equations, especially the last few equations, requires perseverance in making sense of problems and of representations (MP1).

Ask students to share their responses and reasoning for the last four questions. After each student explains, ask the class if they agree or disagree and discuss any disagreements.

Make sure students see that in cases such as and , the solutions to each equation may not necessarily be opposites, as was the case in the preceding equations. For example, in the last question, we want to find a number that produces 4 when it is squared. That number can be 2 or -2. If the number is 2, then is 3. If the number is -2, then is -1.

If time permits, discuss questions such as:

• “What do you notice about the quadratic equations that have only 1 solution?” (They seem to be written as a factor squared equal to 0.)
• "In an equation like , how can we tell that there are two solutions?" (There are two factors—either of which could make the product 0.)

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).