Previously, students have seen that some quadratic functions have no zeros and that some quadratic equations have no solutions. In this Warm-up, they recall that when a solution is a square root of a negative number, the equation has no solutions. Note that students don’t yet know about any numbers that aren’t real at this point, so it is unnecessary to specify “no real solutions.” (To students, the word “real” would seem like an extra word added for no reason.)

Select students to share their responses and reasoning.

If not mentioned in students’ explanations, point out that we can look at and reason about the structure of these equations to tell which ones have no solutions, without taking any steps to solve. For example, the equation means “a square plus 9 equals 0.” Here are two ways to reason about its structure:

• For two numbers to add up to 0, they have to be opposites. Since the number 9 is positive, the other number must be negative, which a square cannot be.
• A square is always positive, so a square plus a positive number must also be positive and can never equal zero.

These lines of reasoning also allow us to see that has no solutions.

By now, students are pretty familiar with quadratic functions that model the height of objects as a function of time after being launched up or being dropped. They have found the zeros of such functions graphically, by identifying horizontal intercepts, as well as algebraically, by applying the zero product property.

Students have also used graphs to estimate input values that yield nonzero output values. Prior to learning about completing the square or the quadratic formula, however, they did not have the tools to solve for such inputs algebraically. In this activity, students apply the knowledge and skills they recently developed to solve contextual problems that they couldn’t previously solve without graphing.

Because algebraic reasoning is the aim of this activity, graphing technology should not be used to solve the equations. It could be used to verify solutions during the Activity Synthesis, however.

If time is limited, assign the first question to half of the class and the second question to the other half. Alternatively, ask students to choose one question to answer.

In this activity, students return to the framing problem they encountered when starting the unit. In that first lesson, they were challenged to use an entire sheet of paper to frame a picture and ensure that the thickness was uniform all around. At that time, students did not have adequate knowledge to solve the problem methodically, so they relied mainly on guessing and checking. Since then, students have developed their understanding about writing and solving quadratic equations. They are now ready to formulate the problem effectively and solve it methodically.

Students must reason abstractly and quantitatively to interpret the equations in the situation (MP2).

Because algebraic reasoning is the aim of this activity, graphing technology should not be used to solve the equations. It could be used to verify solutions during the Activity Synthesis, however.