Lesson 3

Solving Quadratic Equations by Reasoning

Lesson Narrative

In this lesson, students begin to solve quadratic equations by reasoning about what values would make the equations true and by using structure in the equations. The idea that some quadratic equations have two solutions is also made explicit. Students may begin to record their reasoning process as steps for solving, but this is not critical at this point as it will be emphasized in a later lesson.

As students reason about an equation, they may intuitively perform the same operation on each side of the equal sign to get closer to the solution(s). When they reach equations of the form \(x^2=\text{some number}\), it is important to refrain from telling students to “take the square root of each side.” Instead, focus on reasoning about values that would make the equation true (MP2).

For example, to solve \(4x^2=100\), we could divide each side by 4 and get \(x^2=25\). Encourage students to interpret this equation as: “Some number being squared gives 25” and to reason: “There are two different values that can be squared to get 25: -5 and 5.”

Reasoning this way helps to curb two common misconceptions:

  • that the only solution to an equation such as \(x^2=25\) is \(\sqrt{25}\). Each positive number has two square roots, one positive and the other negative. By convention, the radical symbol \(\sqrt{\phantom{3}}\) refers to the positive square root. So the number \(\sqrt{25}\) refers only to the positive square root of 25 and does not capture the negative square root.
  • that squaring is invertible. The inverse of an operation undoes that operation. Suppose we multiply a number by 8. Dividing the product by 8 takes us back to the original number, so we say that division by 8 is the inverse operation of multiplication by 8, and that multiplication by 8 is invertible.

Suppose we square -3, which gives 9. The operation of taking the square root using radical symbol takes 9 to \(\sqrt 9\), which is positive 3, not the original number. Because there are two possible numbers whose square is 9, we don’t consider squaring to be invertible.

When solving an equation such as \(x^2=49\), these notations are commonly used to express the solutions:

  • \(x = 7\) or \(x = \text-7\)
  • \(x = 7\) and \(x = \text-7\)

The use of “or” is really a shorthand for: "If \(x\) is a number such that \(x^2=49\), then \(x=7\) or \(x=\text-7\).” The use of “and” is a shorthand for: “Both \(x=7\) and \(x=\text-7\) are values that make the equation \(x^2=49\) true." Either notation can be appropriate, depending on how the question is stated.

Technology isn't required for this lesson, but there are opportunities for students to choose to use appropriate technology to solve problems. Consider making technology available.

Solving the problems in the lesson gives students many opportunities to engage in sense making, perseverance, and abstract reasoning (MP1, MP2).

Learning Goals

Teacher Facing

  • Find the solutions to simple quadratic equations and justify (orally) the reasoning that leads to the solutions.
  • Understand that a quadratic equation may have two solutions.

Student Facing

  • Let’s find solutions to quadratic equations.

Learning Targets

Student Facing

  • I can find solutions to quadratic equations by reasoning about the values that make the equation true.
  • I know that quadratic equations may have two solutions.

CCSS Standards

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