# Lesson 16

Working with Quadratics

These materials, when encountered before Algebra 1, Unit 7, Lesson 16 support success in that lesson.

## 16.1: Order of Operations and Roots (10 minutes)

### Warm-up

In this warm-up, students use the order of operations to find the value of expressions involving square roots. In the associated Algebra 1 lesson, students will use the quadratic formula, so understanding the order of operations will be useful.

### Student Facing

Find the value of these expressions.

- \(\sqrt{9}+2\)
- \(\frac{\sqrt{16}}{2}\)
- \((\sqrt{25})^2+6.2)\)
- \((\frac{\sqrt{100}}{4} - \frac{\sqrt{64}}{2})\)
- \(\sqrt{1+ 15}\)
- \(\sqrt{4^2 + 3^2}\)

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

The purpose of the discussion is to remind students how square roots fit into the order of operations. Ask students,

- “Why is the last expression not equal to 7?” (The expression under the square root acts like an understood parentheses, so the values are squared and summed before the square root applies.)
- “Which is a greater value: \(\sqrt{4}+5\) or \(\sqrt{4+5}\)? Explain how you know.” (\(\sqrt{4}+5\) is greater, because the square root only applies to the 4 rather than the sum of 4 and 5. \(\sqrt{4}+5 = 7\) and \(\sqrt{4+5}=3\))

## 16.2: Finding Coefficients (15 minutes)

### Activity

In this activity students rewrite quadratic equations into the form \(ax^2 + bx + c = 0\), then identify the values for \(a, b,\) and \(c\), then use those values to find \(b^2 - 4ac\). In the associated Algebra 1 lesson, students work with the quadratic formula. Identifying the values to be used in the quadratic formula and using the values to find the discriminant supports students to use the quadratic formula in its entirety.

### Launch

Ask students to describe a coefficient, then identify the coefficient of the linear term and the constant in the expressions.

- \(2x^2 + 3x + 4\) (coefficient of the linear term: 3, constant term: 4)
- \(x - 2\). (coefficient of the linear term: 1, constant term: -2)

Tell students, “For an upcoming lesson, it will be useful to identify the coefficients of a quadratic equation. First, the equation must be in standard form and equal to zero, like \(ax^2 + bx + c = 0\). Then, the coefficient of the quadratic term is \(a\), the coefficient of the linear term is \(b\), and the constant term is \(c\).”

### Student Facing

Rewrite the equation in standard form \(ax^2 + bx + c = 0\), then identify \(a, b,\) and \(c\). Then compute \(b^2 - 4ac\).

- \(x^2 - 3x + 5 = 0\)
- \(3x^2 - 4 + x = 0\)
- \(\text{-}2x^2 + 5x = 11\)
- \(3x^2 + 5x = 9 - 4x\)
- \(\frac{2x^2}{3} + 6x -13 = 13\)
- \(x^2 - 9 = 0\)
- \(9+x-4x^2 = 1\)
- \((x+2)(x-3) = 0\)

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

The purpose of the discussion is to recognize any preliminary work that needs to be done before using the quadratic formula. Select students to share their solutions. Ask students,

- “For the first question, Noah says that \(a = \text{-}1, b = 3\), and \(c = \text{-}5\). His paper is given full credit. Explain why those values are correct.” (Subtracting or adding each term to both sides results in the equation \(0 = \text{-}x^2 + 3x - 5\), which means Noah’s values are correct.)
- “Before selecting \(a, b,\) and \(c\), what needs to be done to the equation to find the values accurately?” (The equation needs to be rewritten so that one side of the equation is zero and the other side is in standard form.)
- “When you find \(b^2 - 4ac\), what are the first things you need to do for order of operations?” (Squaring the value for \(b\) and multiplying 4, \(a\), and \(c\) should be done before subtracting.)

## 16.3: Practicing Methods for Solving Quadratic Equations (15 minutes)

### Activity

In this activity, students solve quadratic equations using any method they choose. In the associated Algebra 1 lesson, students are introduced to the quadratic formula to solve quadratic equations. Here, students are reminded the other methods can be more or less efficient for solving equations.

### Launch

Ask students what methods they have for solving quadratic equations. List them in a place for all to see. If any of these methods are not mentioned, remind students of the options:

- Factor and use the zero product property.
- Get the equation into the form \((x+b)^2 = b\).
- Complete the square.

### Student Facing

Solve each of these quadratic equations by either rewriting the expression in factored form or completing the square. Explain or show your reasoning for the method you choose to use.

- \(x^2 - 3x - 4 = 0\)
- \(x^2 + x = 6\)
- \(x^2 + 6x + 7 = 5\)
- \(x^2 +12 = 7x\)
- \(x^2 + 3x - 5 = 0\)

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

The purpose of the discussion is to evaluate the methods students already know for solving quadratic equations. Select students to share their solutions. After each solution, ask if any students used another method for solving the question. Ask students about the benefits and difficulties for using each of the methods.