# Lesson 18

Solving Quadratics

These materials, when encountered before Algebra 1, Unit 7, Lesson 18 support success in that lesson.

## 18.1: Math Talk: Operations with Roots (5 minutes)

### Warm-up

The purpose of this Math Talk is to elicit strategies and understandings students have for order of operations involving square roots. 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 find values using the quadratic formula.

### 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

Evaluate mentally:

\(\sqrt{100}-15\)

\(\sqrt{125-10^2}\)

\(20-2\sqrt{49}\)

\(\sqrt{4^2+3^2}\)

### Student Response

For access, consult one of our IM Certified Partners.

### 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?”

## 18.2: Checking Brother’s Work (20 minutes)

### Activity

In this activity, students are asked about ways to check solutions for equations and apply the zero product property to solve a quadratic equation in a new context. In the associated Algebra 1 lesson, students examine ways to check their work when using the quadratic formula. This activity supports students to use additional methods to check their solutions when solving equations.

### Student Facing

Priya's older brother is working on some higher-level math work and claims that \(x = 3\) is a solution to the equation \(x^3 - 5x^2 -2x = \text{-}24\).

- Explain how she could check that his solution is correct using each of these tools.
- A basic calculator
- A graphing tool

- When looking at his work, Priya sees that he has the equation \((x-3)(x^2 -2x - 8) = 0\). Knowing the zero product property holds, Priya recognizes that this equation means \(x-3 = 0\) or \(x^2 -2x - 8 = 0\) for this question. Find 2 other solutions to the original equation. Explain or show your reasoning.

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

The purpose of the discussion is to give students methods for checking their work. Select students to share their solutions and methods. For the second question, ask students what methods they used to solve the quadratic equation. Ask students,

- “What does the zero product property say and why does it hold for the equation in Priya's brother’s work?” (The zero product property says that when two things are multiplied to get zero, at least one of the things must be zero. From Priya's brother’s work, he has that 2 things multiply to make zero: \(x+3\) and \(x^2 - 2x - 8\), so one of those must be zero.)
- “How can you check that your solutions, -2 and 4, work for the original equation?” (Substitute the values into the original equation or look at a graph.)

## 18.3: Steps to Using the Quadratic Formula (15 minutes)

### Activity

In this activity, students list some of the steps in the process of using the quadratic formula to solve a quadratic equation. In the associated Algebra 1 lesson, students examine some common errors when using the quadratic formula. This work supports students to be reminded of the order of operations as well as determining which values to use in the quadratic formula.

### Student Facing

The quadratic formula solves equations of the form \(ax^2 + bx + c = 0\) using the equation \(x=\frac{\text{-}b \pm \sqrt{b^2 - 4ac}}{2a}\).

Andre wants to use the quadratic formula to solve \(x^2 - 7x = \text{-}12\).

- What should Andre do first?
- What values of \(a, b,\) and \(c\) should he use?
- After substituting the values into the quadratic formula, what is the order he should use to calculate the solutions?
- Use the quadratic formula to solve the equation.
- Check your solutions.

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

The purpose of the discussion is to remind students of the order in which things should be done when using the quadratic formula. Select students to share their solutions.

As students share the order to work on the calculation, it may be helpful to create a list for all to see.

- Ensure the equation is in the form \(ax^2 + bx + c = 0\).
- Determine \(a, b, c\) from the equation.
- Substitute the values into the quadratic formula (it may help to put each value in parentheses).
- Multiply all terms in the quadratic formula: \(\text{-}b, b^2, 4ac, 2a\).
- Subtract the values inside the square root.
- Find the square root of the value.
- Write as 2 separate solutions, one using addition and one subtraction based on the \(\pm\).
- Add and subtract in the numerator for each solution.
- Divide the numerator by the denominator.
- Check your work (substitute the values into the original equation or graph).