# Lesson 19

Real and Non-Real Solutions

## 19.1: Notice and Wonder: Where Is It 0? (5 minutes)

### Warm-up

The purpose of this warm-up is to elicit the idea that although it may be difficult to tell from a function's expression whether it will equal 0 for any real values of \(x\), it is relatively straightforward to tell this from looking at the function's graph. Connections students make between a function's coefficients and its graph will be useful when students create their own equations that have real or non-real solutions in a later activity. While students may notice and wonder many things about these graphs and their intercepts, connections between the coefficients of the functions and the locations of the intercepts of their graphs are the important discussion points.

This prompt gives students opportunities to see and make use of structure (MP7). The specific structure they might notice is the effect on the graph of changing the constant term, and the effect this has on the number of real \(x\)-values at which the function is 0. Students will explore the effect of the other terms in later activities.

### Launch

Display the graphs and functions from the task statement for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice and wonder with their partner, followed by a whole-class discussion.

### Student Facing

What do you notice? What do you wonder?

\(f(x)=2x^2+4x\)

\(f(x)=0\) when \(x=0,\text{-}2\).

\(g(x)=2x^2+4x+2\)

\(g(x)=0\) when \(x=\text-1\).

\(h(x)=2x^2+4x+4\)

\(h(x)=0\) when \(x=\text-1 + i , \text-1 - i\).

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the graphs. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information.

An important point of the discussion is the pattern that as the constant term increases, the graph moves up, which affects the number of real zeros of the function. If this does not come up during the conversation, ask students to discuss this idea. Students will study translations and other transformations of functions in greater depth in a future unit.

## 19.2: Real or Not? (15 minutes)

### Optional activity

This activity is optional because it includes additional practice students may not need. In this activity, students predict which equations will have real solutions and which ones will not.

Press students to give reasons for their predictions rather than guessing. There are many valid methods of making a prediction about whether there are real solutions. Monitor for students who use the following strategies to share during the whole-class discussion:

- try to factor the expression and conclude that the solutions are probably not real if it seems impossible to factor
- sketch a graph of the function to see if it has \(x\)-intercepts
- use the quadratic formula or completing the square to find the actual solutions
- strategically use the quadratic formula to determine whether the discriminant, \(b^2-4ac\), is negative and conclude that the solutions will be real only if it's not negative

In the following activity, it will be helpful for students to have a reliable and efficient method for generating quadratic functions that have real or non-real zeros. Factoring is inefficient and it would be difficult for students to use it to generate functions that have non-real zeros. Looking at the function's graph is a more reliable way to see whether it has real zeros, but it's not easy to use to purposely create a function that will or will not have real zeros. Checking the sign of the discriminant is a comparatively quick and reliable test for whether a function will have real zeros, and can also be used to create a function that will or will not have real zeros.

### Launch

Arrange students in groups of 2. After students make their predictions, display the solutions for students to check their predictions or, if time allows, ask students to solve the equations to check their predictions.

*Representation: Internalize Comprehension.*Activate or supply background knowledge. For example, ask students to consider the different methods they have used to find the solutions to quadratic functions.

*Supports accessibility for: Memory; Conceptual processing*

### Student Facing

Here are some equations:

Equation | Prediction |
---|---|

\(x^2-6x+5=0\) | \(\hspace{1.5in}\) |

\(x^2-6x+13=0\) | |

\(\text-x^2+6x-9=0\) | |

\(\text-x^2-9=0\) |

- Which equations will have real solutions, and which ones will not? Write your prediction in the table.

Pause here for discussion. - What advice would you give to someone who is trying to figure out whether a quadratic equation has real solutions?

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

Select previously identified students to share in the order given in the narrative. Here are some important points to highlight during discussion:

- The first question only asked for a prediction about whether the solution(s) will be real, and not for the actual solution.
- Factoring and graphing are less efficient than calculating \(b^2-4ac\).
- The ideas behind each of these strategies can be used to write a quadratic function with real zeros, or one that does not have real zeros.
- Strategically using \(b^2-4ac\) is an efficient way to create a quadratic function with a specific type of zeros by picking values for \(a\), \(b\), and \(c\) that make the value of this expression positive (real) or negative (non-real).

If students ask whether there is a special name for \(b^2-4ac\), this is an appropriate time to tell them it's called the *discriminant*.

*Speaking: MLR8 Discussion Supports.*Use this routine to support whole-class discussion. As students share their strategies for figuring out whether a quadratic equation has real solutions, ask students to restate what they heard using precise mathematical language. Consider providing students time to restate what they hear to a partner before selecting one or two students to share with the class. Ask the original speaker if their peer was accurately able to restate their thinking. Call students’ attention to any words or phrases that helped clarify the original statement such as, “the expression under the radical in the quadratic formula.” This provides more students with an opportunity to produce language as they interpret the reasoning of others.

*Design Principle(s): Support sense-making*

## 19.3: Make Your Own (15 minutes)

### Optional activity

This activity is optional because it includes additional practice students may not need. The goal of this activity is for students to use the strategies discussed in the previous activity to build their own quadratic equations with either real or non-real solutions. While guess-and-check is possible, the emphasis here should be on strategic use of the quadratic formula or completing the square. For example, in a previous lesson where students completed the square to find non-real solutions to quadratic equations, they saw that a quadratic such as \((x-4)^2=k\) has non-real solutions when \(k<0\).

Monitor for students who use the strategy of choosing \(a\), \(b\), and \(c\) so that \(b^2-4ac\) will be either negative or not and for students using ideas from completing the square.

### Launch

Arrange students in groups of 2. They should work with their partner to create the quadratic equations, then solve the equations on their own. Once both partners have found solutions, they should compare answers and work to reach agreement if the answers aren’t equivalent.

If time, pairs can also swap equations and solve each other's, after first predicting whether each equation will have real or non-real solutions.

*Reading, Writing, Speaking: MLR3 Clarify, Critique, Correct.*Before students share their observations about the presented equations, present an ambiguous response. For example, “The number on the left side is too big so the solution will be complex.” Ask students to identify the error, critique the reasoning, and write a correct explanation. As students discuss with a partner, listen for students who identify and clarify the ambiguous language in the statement. Invite students to share their critiques and corrected explanations with the class. Listen for and amplify the language students use to explain how they predict whether the number under the square root is positive or negative. This helps students evaluate, and improve upon, the written mathematical arguments of others, as they identify patterns in equations with real or non-real solutions.

*Design Principle(s): Optimize output (for explanation); Maximize meta-awareness*

*Action and Expression: Internalize Executive Functions.*To support development of organizational skills, check in with students within the first 2–3 minutes of work time. Look for students who are using the quadratic formula or completing the square to create and solve quadratic equations.

*Supports accessibility for: Memory; Organization*

### Student Facing

- Create three different quadratic equations. At least one should have solutions with an imaginary part, and at least one should have solutions with only real parts.
- Solve your equations. Write down your equations and their solutions in the table.
Equation Solutions \(\hspace{3in}\) \(\hspace{2in}\)

### Student Response

For access, consult one of our IM Certified Partners.

### Anticipated Misconceptions

Students who are not yet comfortable using the discriminant or other algebraic techniques to create their equations may benefit from using graphing technology to experiment with different values of \(a\), \(b\), and \(c\) until they get a graph that either crosses the \(x\)-axis or doesn't. Encourage these students to find the discriminant of their equation to confirm that it has the type of solutions they're looking for. Consider encouraging them to change one of the values to make the discriminant change sign, and then graph the resulting equation to see how it's different from the one they started with.

### Activity Synthesis

Invite 2-3 previously identified pairs to share one of their equations, how they knew what type of solutions the equation would have, and how they found the solutions.

Then invite students to share observations about how the equations with real solutions are different from the ones with non-real solutions. Depending on the specific equations students wrote, patterns may or may not be apparent. This is why it's helpful to have a quick test like checking the sign of the discriminant to figure out whether a quadratic equation will have real solutions.

## Lesson Synthesis

### Lesson Synthesis

Use the applet to display the graphs of the equations written by students. The applet will show the number of real and non-real solutions. (Note that setting \(a=0\) will not produce a quadratic function.)

Ask students for an example of a function they found that had only real zeros. By either moving the sliders or typing in the numbers directly as values of \(a\), \(b\), and \(c\), change the graph so that it represents the example function. Ask students how we could change \(a\), \(b\), or \(c\) to make this function have non-real zeros, and then change the value based on their suggestions. Press students for reasons to make each of the changes they suggest. For example, a student might suggest increasing the value of \(c\), and the reason might be that it would raise the graph above the \(x\)-axis.

Once several changes have been made to the graph, ask students for an example of a function they found that had only non-real solutions, and ask how we could change it so that it has real solutions.

## 19.4: Cool-down - Make it Complex (5 minutes)

### Cool-Down

For access, consult one of our IM Certified Partners.

## Student Lesson Summary

### Student Facing

By looking at the graph of a quadratic function, we can tell whether it has real zeros or not. The real zeros are the \(x\)-coordinates of the points where the graph touches the \(x\)-axis. So, if the graph doesn't touch the \(x\)-axis at all, then the function does not have real zeros. If that happens, we won't be able to tell exactly what the zeros are just by looking at the graph, but we will be able to find them by setting the function equal to 0 and solving algebraically.

We can also tell whether \(y=ax^2+bx+c\) has real zeros by using the quadratic formula strategically, since the quadratic formula tells us the \(x\)-values that make a quadratic function be 0. For reference, here is the quadratic formula:

\(\displaystyle x=\dfrac{-b \pm \sqrt{b^2-4ac}}{2a}\)

Imaginary numbers are the square roots of negative numbers, so it will be helpful to look at the part of the quadratic formula that involves a square root, which is \(\sqrt{b^2-4ac}\). If \(4ac\) is greater than \(b^2\), then the number under the radical will be negative, which means that the zeros of the quadratic function will have an imaginary part.

For example, consider the quadratic function \(y=x^2+2x+2\). How can we tell if it has real zeros?

One way is to look at the graph, shown here. Since its graph doesn't intersect the \(x\)-axis, this function doesn't have any real zeros.

We can also tell by checking whether \(4ac\) is greater than \(b^2\). Since \(a=1\), \(b=2\), and \(c=2\), \(b^2-4ac=\text-4\), so the zeros of this function have an imaginary part. We can find out exactly what the zeros are by using the quadratic formula or completing the square.