Exploring Roots of a Quadratic Equation, Page 3
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 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
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
- 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?
- CHALLENGE: Can you find another parabola and identify the roots?
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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Math Forum Resources
- Analytic Geometry Formulas
- Analytic geometry (a branch of geometry in which points are represented with respect to a coordinate system, such as cartesian coordinates) formulas for figures in one, two, and three dimensions: points, directions, lines, triangles, polygons, conic sections, general quadratic equations, spheres, etc.
- Describing the Graph of an Equation
- Which of the following is true of the graph of the equation: y = 2x^2 - 5x + 3?
- Equations and Factoring
- Solve. Identify all double roots: 2(r^2 + 1)=5r
- Graphing Parabolas
- How do you know how to graph a parabola from looking at its equation?
- Learning to Factor
- A sampling of answers from our archives.
- What is a Quadratic Equation?
- What is a quadratic equation? What is it used for? How can we use it to solve everyday problems?
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