Using Quadratic Equations to Model Situations and Solve Problems

10 min

Narrative

This Warm-up activates some familiar skills for writing and solving equations, which will be useful for specific tasks throughout the lesson.

It may have been a while since students thought about writing an equation for a line passing through two points. The two questions here are intentionally quite straightforward. Monitor for students taking different approaches, such as:

  • Plotting the points and determining the slope and yy-intercept of a line passing through the points.
  • Computing the slope by finding the quotient of the difference between the yy-coordinates and the difference between the xx-coordinates.
  • Considering which operation on each xx-coordinate would produce its corresponding yy-coordinate.

Students have not yet solved a quadratic equation like the one in the second question, but they have learned and extensively practiced the skills needed to solve it. The two main anticipated approaches are:

  • Reasoning algebraically, by performing the same operation to each side of the equation, applying the distributive property to expand factored expressions, combining like terms, rewriting an expression in factored form, and applying the zero product property
  • Graphing y=x+1y=x+1 and y=(x−2)2−3y=(x-2)^2-3 and observing the xx-coordinate of each point of intersection

Launch

Give students 5–7 minutes to work on the questions, then follow with a whole-class discussion.

Student Task

  1. Write an equation representing the line that passes through each pair of points.

    1. (3,3)(3,3) and (5,5)(5,5)
    2. (0,4)(0,4) and (-4,0)(\text-4,0)
  2. Solve this equation: x+1=(x−2)2−3x+1=(x-2)^2-3. Show your reasoning.

Sample Response

  1. Equations (or their equivalents):
    1. y=xy=x
    2. y=x+4y=x+4
  2. x=0x=0 or x=5x=5. Sample reasoning:

    \displaystyle \begin {align} x+1 &=(x-2)^2-3 \\ x+1 &= x^2 -4x+4 - 3\\ x+1 &= x^2 -4x +1\\ x &= x^2-4x\\0 &= x^2 -5x\\ 0&=x(x-5)\\ x = 0 & \quad \text {or} \quad x=5 \end{align}

Activity Synthesis (Teacher Notes)

Invite students taking different approaches to share their work. Ensure that students see more than one way to think about the equation representing a line for the first question and recognize that the second equation can be solved algebraically.

Standards
Building On
  • F-LE.2·Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
  • F-LE.2·Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
  • F-LE.2·Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
  • HSF-LE.A.2·Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
Addressing
  • A-REI.4.b·Solve quadratic equations by inspection (e.g., for x² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.
  • A-REI.4.b·Solve quadratic equations by inspection (e.g., for x² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.
  • A-REI.4.b·Solve quadratic equations by inspection (e.g., for x² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.
  • HSA-REI.B.4.b·Solve quadratic equations by inspection (e.g., for <span class="math">\(x^2 = 49\)</span>), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as <span class="math">\(a \pm bi\)</span> for real numbers <span class="math">\(a\)</span> and <span class="math">\(b\)</span>.

15 min

10 min