# Lesson 13

Constants in Quadratic Equations

These materials, when encountered before Algebra 1, Unit 7, Lesson 13 support success in that lesson.

## 13.1: Math Talk: Halved and Squared (5 minutes)

### Warm-up

The purpose of this Math Talk is to elicit strategies and understandings students have for finding a value that needs to be added to a quadratic expression to complete the square. These understandings help students develop fluency and will be helpful later in the associated Algebra 1 lesson when students will need to be able to complete the square.

### Launch

Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

### Student Facing

For each value of \(b\), mentally find \(\left(\frac{b}{2} \right)^2\).

\(b = 6\)

\(b = \frac{1}{2}\)

\(b = \frac{2}{5}\)

\(b = 0.8\)

### Student Response

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### Activity Synthesis

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

- “Who can restate \(\underline{\hspace{.5in}}\)’s reasoning in a different way?”
- “Did anyone have the same strategy but would explain it differently?”
- “Did anyone solve the problem in a different way?”
- “Does anyone want to add on to \(\underline{\hspace{.5in}}\)’s strategy?”
- “Do you agree or disagree? Why?”

## 13.2: Solving Quadratics with Perfect Squares (15 minutes)

### Activity

In this activity, students solve equations of the form \((x+a)^2 = b\). In the associated Algebra 1 lesson, students solve equations using completing the square. After they complete the square, students can use the methods from this activity to solve the equation.

### Student Facing

Solve each of these equations for all values of \(x\) that make the equation true.

- \((x+2)^2 = 9\)
- \((x-\frac{1}{2})^2 = 4\)
- \((x+1)^2 = 8 + 1\)
- \((x-\frac{1}{3})^2 = \frac{10}{9}- \frac{1}{9}\)
- \((x-6)(x-6) = 81\)

### Student Response

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### Activity Synthesis

The purpose of the discussion is to clarify that the methods used in this activity are specific to equations that are in a certain form.

Remind students that this method only works when the equation is of the form \((x+a)^2 = b\). For example, these equations would require other methods for solving:

- \((x+2)(x-5)=0\) (\(x = \text{-}2, x = 5\))
- \((x+1)(x-1) = 24\) (\(x = 5, x = \text{-}5\))
- \((x-3)(x+12) = 9x\) (\(x = 6, x = \text{-}6\))

Ask students what methods they could use to solve the equations here.

## 13.3: Make It a Perfect Square (20 minutes)

### Activity

In this activity, students use the patterns they noticed about quadratic expressions in standard form to add a constant value to the quadratic and linear term to make it a perfect square. After adding the value, students rewrite the expression in factored form. In the associated Algebra 1 lesson, students work to complete the square to solve equations. The work of this activity helps students develop some fluency with adding the correct constant as part of the process for completing the square.

### Student Facing

For each expression:

- Find a value that could be added as a constant term to make each expression a perfect square.
- Add the value you found and rewrite the expression in factored form.

- \(x^2 + 20x\)
- \(x^2 - 4x\)
- \(x^2 - 2x\)
- \(x^2 + x\)
- \(x^2 + 5x\)
- \(x^2 + 1.4x\)

### Student Response

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### Activity Synthesis

The purpose of the discussion is to recognize methods for adding a constant to an expression to complete the square. Select students to share their solutions and methods for solving the questions. Ask students,

- “How did you find the value for the constant you added?” (I know the number must be a perfect square so that twice the square root of the number is the linear coefficient. For example, in the first problem it is 100 since \(\sqrt{100} = 10\) and \(2 \boldcdot 10 = 20\).)
- “How do you know that the second expression has factors \((x-2)^2\) rather than \((x+2)^2\)?” (Since the linear term is negative, the factors will involve subtraction.)
- “Is the constant value added always positive? Explain or show your reasoning.” (Yes, it is always positive. Since the constant term comes from squaring the constant value of the factors and squares are always positive, the value you add is always positive.)