Solving Quadratic Equations by Using Factored Form

10 min

Narrative

In this activity, students find at least one solution of x2−2x−35=0x^2-2x-35=0 by substituting different values of xx, evaluating the expression, and checking if it has a value of zero. Experiencing this inefficient method puts students in a better position to appreciate why it may be desirable to write x2−2x−35x^2-2x-35 in factored form and use the zero product property.

Launch

Once students have had a chance to evaluate the expression x2−2x−35x^2-2x-35, using their chosen number for xx, ask if anyone found a value that made the expression equal 0. (It's 7.) Give students a couple of minutes to look for the other value that makes the expression equal 0.

Student Task

Let's try to find at least one solution to x2−2x−35=0x^2-2x-35=0.

  1. Choose a whole number between 0 and 10.
  2. Evaluate the expression x2−2x−35x^2-2x-35, using your number for xx.
  3. If your number doesn't give a value of 0, look for someone in your class who may have chosen a number that does make the expression equal 0. Which number is it?
  4. There is another number that would make the expression x2−2x−35x^2-2x-35 equal 0. Can you find it?

Sample Response

  1. Answers vary.
  2. Here are all the possibilities:
    xx 0 1 2 3 4 5 6 7 8 9 10
    x2−2x−35x^2-2x-35 -35 -36 -35 -32 -27 -20 -11 0 13 28 45
  3. 7
  4. It's -5, but it is possible that no students will consider negative numbers.
Activity Synthesis (Teacher Notes)

If a student found that -5 makes the expression equal 0, ask them to demonstrate that (-5)2−2(-5)−35(\text-5)^2-2(\text-5)-35 equals 0.

Discuss with students:

  • "Can x2−2x−35x^2-2x-35 be written in factored form? What are the factors?" (Yes, (x−7)(x-7) and (x+5)(x+5).)
  • "If x2−2x−35x^2-2x-35 can be written as (x−7)(x+5)(x-7)(x+5), can we solve (x−7)(x+5)=0(x-7)(x+5)=0 instead?" (Yes.) "Will the solutions change if we use this equation?" (No. The equations are equivalent, so they have the same solutions.)
  • "Why might someone choose to rewrite x2−2x−35=0x^2-2x-35=0 and solve (x−7)(x+5)=0(x-7)(x+5)=0 instead?" (Because the expression is equal to 0, rewriting it in factored form allows us to use the zero product property to find both solutions. It may be more efficient than substituting and evaluating many values for xx. It also makes it possible to see how many solutions there are, which is not always easy to tell when the quadratic expression is in standard form.)
Standards
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

10 min