# Lesson 19

Quadratic Steps

- Let’s explore steps for solving an equation.

### 19.1: Quadratic Error

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.

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

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

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

### 19.2: Multiplying to Make Perfect Squares

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?
- \(a\)
- \(3a\)
- \(4a\)
- \(6a\)
- \(9a\)

### 19.3: Stepping Through Completing the Square

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

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

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