Applying the Quadratic Formula (Part 1)

5 min

Narrative

Previously, students have seen that some quadratic functions have no zeros and that some quadratic equations have no solutions. In this Warm-up, they recall that when a solution is a square root of a negative number, the equation has no solutions. Note that students don’t yet know about any numbers that aren’t real at this point, so it is unnecessary to specify “no real solutions.” (To students, the word “real” would seem like an extra word added for no reason.)

Launch

Arrange students in groups of 2. Give students quiet think time and then time to share their thinking with a partner.

Student Task

Here is an example of someone solving a quadratic equation that has no solutions:

\displaystyle \begin {align} (x+3)^2+9 &=0\\ (x+3)^2 &=\text-9\\ x+3 &=\pm \sqrt{\text-9} \end {align}

  1. Study the example. At what point did you realize the equation had no solutions?
  2. Explain how you know the equation 49+x2=049+x^2=0 has no solutions.

Sample Response

  1. Sample responses:
    • in the second line, when noticing that the squared quantity equals a negative number
    • in the third line, when seeing the square root of a negative number
  2. Sample response: Solving this equation involves rearranging the terms, which leads to x2x^2 on one side and a negative number on the other. It is not possible for the square of a number to be negative.
Activity Synthesis (Teacher Notes)

Select students to share their responses and reasoning.

If not mentioned in students’ explanations, point out that we can look at and reason about the structure of these equations to tell which ones have no solutions, without taking any steps to solve. For example, the equation (x+3)2+9=0(x+3)^2 + 9=0 means “a square plus 9 equals 0.” Here are two ways to reason about its structure:

  • For two numbers to add up to 0, they have to be opposites. Since the number 9 is positive, the other number must be negative, which a square cannot be.
  • A square is always positive, so a square plus a positive number must also be positive and can never equal zero.

These lines of reasoning also allow us to see that 49+x2=049+x^2=0 has no solutions.

MLR8 Discussion Supports. Provide students with the opportunity to rehearse what they will say with a partner before they share with the whole class.
Advances: Speaking
Standards
Addressing
  • A-REI.A·Understand solving equations as a process of reasoning and explain the reasoning
  • HSA-REI.A·Understand solving equations as a process of reasoning and explain the reasoning.
Building Toward
  • A-REI.4·Solve quadratic equations in one variable.
  • A-REI.4·Solve quadratic equations in one variable.
  • A-REI.4·Solve quadratic equations in one variable.
  • HSA-REI.B.4·Solve quadratic equations in one variable.

15 min

15 min