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