Lesson 2

When and Why Do We Write Quadratic Equations?

Lesson Narrative

In this lesson, students revisit some situations that can be modeled with quadratic functions. They analyze and interpret given equations, write equations to represent relationships and constraints (MP4), and work to solve these equations. In doing so, students see that sometimes solutions to quadratic equations cannot be easily or precisely found by graphing or reasoning.

In an earlier unit, students saw that when a function is defined by a quadratic expression in factored form, the zeros of the function could be easily identified. Here, they notice that when a quadratic equation is written as \(\text{factored form}=0\), solving the equation is also relatively simple. This revelation motivates upcoming work on rewriting quadratic expressions into factored form.

Learning Goals

Teacher Facing

  • Recognize the limitations of certain strategies used to solve a quadratic equation.
  • Understand that the factored form of a quadratic expression can help us find the zeros of a quadratic function and solve a quadratic equation.
  • Write quadratic equations and reason about their solutions in terms of a situation.

Student Facing

  • Let’s try to solve some quadratic equations.

Required Preparation

Be prepared to display a graph using technology during the activity synthesis of The Flying Potato Again.

Learning Targets

Student Facing

  • I can recognize the factored form of a quadratic expression and know when it can be useful for solving problems.
  • I can use a graph to find the solutions to a quadratic equation but also know its limitations.

CCSS Standards

Building On

Addressing

Building Towards

Glossary Entries

  • factored form (of a quadratic expression)

    A quadratic expression that is written as the product of a constant times two linear factors is said to be in factored form. For example, \(2(x-1)(x+3)\) and \((5x + 2)(3x-1)\) are both in factored form.

  • quadratic equation

    An equation that is equivalent to one of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(a \neq 0\).

  • quadratic expression

    A quadratic expression in \(x\) is one that is equivalent to an expression of the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants and \(a \neq 0\).

  • standard form (of a quadratic expression)

    The standard form of a quadratic expression in \(x\) is \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a\) is not 0.

  • zero (of a function)

    A zero of a function is an input that yields an output of zero. If other words, if \(f(a) = 0\) then \(a\) is a zero of \(f\).