# Lesson 19

Quadratic Steps

These materials, when encountered before Algebra 1, Unit 7, Lesson 19 support success in that lesson.

## 19.1: Quadratic Error (5 minutes)

### Warm-up

In this warm-up, students are asked to spot a common error that often takes the form \(\sqrt{a^2 - b^2} = a - b\). The format of the question here closely resembles the values that students will see when working with the quadratic formula in the associated Algebra 1 lesson.

### Student Facing

Here is Han’s work to solve an equation. Determine the error he made and be prepared to explain the correct way to solve it.

**\(x= \text{-}3 + \sqrt{3^2-4\boldcdot1\boldcdot2}\)**

**\(x= \text{-}3+ 3-2\boldcdot1\boldcdot2\)**

**\(x=\text{-}4\)**

### Student Response

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### Activity Synthesis

The purpose of the discussion is to remind students to be careful about common errors when working with square roots. Ask students,

- “How can you remember where square roots fit into the order of operations?” (I like to think of the expression under the square root as a separate question to evaluate before doing anything else.)
- “Two common errors occur when using squares and square roots. First, \((a+b)^2 = a^2 + b^2\) is not correct in most cases. Next, \(\sqrt{a^2 + b^2} = a + b\) is also not true in most cases. Choose values for \(a\) and \(b\) to assure yourself that these are not true.” (For the first one, \((2 + 3)^2 = 25\) but \(2^2 + 3^2 = 13.\) For the second one, \(\sqrt{2^2 + 3^2} = \sqrt{13}\) which is about 3.6 but \(2+3=5\).)

## 19.2: Multiplying to Make Perfect Squares (15 minutes)

### Activity

In this activity, students examine values that can be multiplied by a number to create a perfect square. In the associated Algebra 1 lesson, students prove the quadratic formula. As part of the process of proving the quadratic formula, students multiply the equation by \(4a\) so that the coefficient of the quadratic term is a perfect square. This work of this activity supports students in understanding that step.

### Student Facing

The class is asked to multiply 5 by a number to make it a perfect square.

- Jada multiplies the number by 5.
- Han multiplies the number by 15.
- Elena multiplies the number by 9.
- Kiran multiplies the number by 20.
- Mai multiplies the number by 45.

- Do you agree with any of the students that their multiplication will make a perfect square?
- Find the pairs of positive integer factors of each of the numbers the students want to use.
- What do you notice about the factors of the values that do create a perfect square? What do you notice about the factors of the values that do not create a perfect square?
- What are some values you could multiply the number 7 by to make it a perfect square?
- If \(a\) is an integer, which of these values could be multiplied by \(a\) so that the product is a perfect square?
- \(a\)
- \(3a\)
- \(4a\)
- \(6a\)
- \(9a\)

### Student Response

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### Activity Synthesis

The purpose of the discussion is to explore what kinds of values can be multiplied by a number to get a perfect square. Select students to share their solutions and things they noticed. Ask students,

- “To complete the square, the coefficient of \(x^2\) must be a perfect square. To make the coefficient of \(x^2\) a perfect square for the equation \(3x^2 - 4x = 5\), what could you multiply the equation by? Explain your reasoning.” (I could multiply by 3, 12, 27, or other values of the form \(3n^2\). That would make the coefficient of \(x^2\) of the form \(9n^2\) which is a perfect square.)
- “A quadratic equation in standard form looks like \(ax^2 + bx + c = 0\). What does that equation look like when you multiply the equation by \(4a\)?” (\(4a^2x^2 + 4ab + 4ac = 0\))

## 19.3: Stepping Through Completing the Square (20 minutes)

### Activity

In this activity, students go through each step of completing the square. In the associated Algebra 1 lesson, students prove the quadratic formula by completing the square with the general form of a quadratic equation in standard form. By closely examining the steps for an actual example, students should be supported to follow the steps for a more abstract example in the associated Algebra 1 lesson.

### Student Facing

For each step of the solution, explain what happened in each step and why that step might be taken.

Solve \(x^2 + 8x - 3 = 6\).

- \(x^2 + 8x = 6 + 3\)
- \(x^2 + 8x + 16 = 9 + 16\)
- \((x+4)^2 = 25\)
- \(x + 4 = \pm 5\)
- \(x = \text{-}4 \pm 5\)
- \(x = 1, \text{-}9\)

### Student Response

Teachers with a valid work email address can click here to register or sign in for free access to Student Response.

### Activity Synthesis

The purpose of the discussion is to understand every step of solving an equation by completing the square. Select students to share their solutions. Ask students,

- “How could the first 2 steps be combined?” (Knowing that the left side should have a constant term of 16, you can add 19 to both sides of the equation right at the beginning.)
- “The way this work is written, the \(\pm\) is kept until the end of the work when the 2 solutions are found. Another option is to write two equations as soon as line 4 of this work. What are the benefits of each method?” (It is less to write if I leave the \(\pm\) until the end. When writing 2 equations, it is easier to remember that there are 2 solutions and the numbers can be combined earlier.)