Exploring Roots of a Quadratic Equation, Page 3

  1. Did you get 2 and -3? The parabola crosses the x-axis at (2, 0) and also at (-3, 0).

    Let's think about those two numbers:
    • The sum of 2 and -3 is -1.
    • The product of 2 and -3 is -6.
  2. 2 and -3 are the roots of the quadratic equation. The roots satisfy the equation:

           y = x^2 + x - 6
           y = (x - 2)(x + 3)
           y = x - 2    or    y = x + 3
           
           If y is 0, then
           0 = x - 2    or    0 = x + 3
           x = 2        or    x = -3
           
           CHECK:
           0 = 2^2 + 2 - 6     or     0 = (-3)^2 + (-3) - 6
           0 = 4 + 2 - 6       or     0 = 9 - 3 - 6
           0 = 6 - 6           or     0 = 9 - 9
           0 = 0               or     0 = 0
           
    
  3. Here's another parabola to graph. Select 0 for a and -4 for b on the blue graph. In other words, click on the point (0, -4) on the blue graph. Now think about what two numbers have sum 0 and product -4? Can you name those two numbers by noting where the parabola crosses the x axis on the green graph?
    • What is the quadratic equation?
    • What is that equation in factored form?
    • What are the roots of the equation?
  4. CHALLENGE: Can you find another parabola and identify the roots?

Final Task

Think about the following:

  • How do a and b affect the location of the parabola?
  • How can you know the quadratic equation by looking at the green graph?

Generalize how the applet can be used to find the roots of a quadratic equation.

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