Math Problem Statement
Find the solution of the equation x2 -2x -3 = 0 by graphing
Solution
To solve the quadratic equation by graphing, we follow these steps:
1. Rewrite the equation in standard form:
The given equation is already in the form: This is the equation of a parabola.
2. Plot the parabola:
To graph the quadratic function, we will calculate a few values of and the corresponding values of , then plot those points.
3. Identify the x-intercepts:
The x-intercepts of the graph (where the graph crosses the x-axis) are the solutions of the equation . These intercepts are the points where .
Let’s start by factoring the quadratic equation: So, the solutions are: These are the x-intercepts, meaning the graph crosses the x-axis at and .
4. Sketch the graph:
- The parabola opens upwards since the coefficient of is positive.
- The x-intercepts are at and .
- The vertex can be found using the formula for the x-coordinate of the vertex, . For this equation, and : Substituting into the equation to find the y-coordinate of the vertex: So, the vertex is at .
By plotting the vertex and the x-intercepts, the parabola will have its minimum point at , and it will pass through and .
Thus, the solution to the equation by graphing is and .
Would you like a graph for better visualization or further details?
5 Questions to Explore Further:
- How do you determine the vertex of a quadratic equation?
- What is the significance of the x-intercepts in a quadratic equation?
- Can all quadratic equations be factored easily? If not, what methods can be used to solve them?
- What does the direction of a parabola (opening upwards or downwards) depend on?
- How can the quadratic formula be used to solve quadratic equations?
Tip:
The vertex of a parabola gives you the minimum or maximum value of the quadratic function, depending on whether the parabola opens upwards or downwards.
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Math Problem Analysis
Mathematical Concepts
Algebra
Quadratic Equations
Graphing
Formulas
Standard form of a quadratic equation: ax^2 + bx + c = 0
Factoring: (x - 3)(x + 1) = 0
Vertex formula: x = -b / 2a
Theorems
Zero Product Property
Vertex of a Parabola Theorem
Suitable Grade Level
Grades 9-10