Math Problem Statement

Find the solution of the equation x2 -2x -3 = 0 by graphing

Solution

To solve the quadratic equation x22x3=0x^2 - 2x - 3 = 0 by graphing, we follow these steps:

1. Rewrite the equation in standard form:

The given equation is already in the form: y=x22x3y = x^2 - 2x - 3 This is the equation of a parabola.

2. Plot the parabola:

To graph the quadratic function, we will calculate a few values of xx and the corresponding values of y=x22x3y = x^2 - 2x - 3, 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 x22x3=0x^2 - 2x - 3 = 0. These intercepts are the points where y=0y = 0.

Let’s start by factoring the quadratic equation: x22x3=(x3)(x+1)=0x^2 - 2x - 3 = (x - 3)(x + 1) = 0 So, the solutions are: x=3orx=1x = 3 \quad \text{or} \quad x = -1 These are the x-intercepts, meaning the graph crosses the x-axis at x=3x = 3 and x=1x = -1.

4. Sketch the graph:

  • The parabola opens upwards since the coefficient of x2x^2 is positive.
  • The x-intercepts are at x=3x = 3 and x=1x = -1.
  • The vertex can be found using the formula for the x-coordinate of the vertex, x=b2ax = -\frac{b}{2a}. For this equation, a=1a = 1 and b=2b = -2: x=22(1)=1x = -\frac{-2}{2(1)} = 1 Substituting x=1x = 1 into the equation to find the y-coordinate of the vertex: y=(1)22(1)3=123=4y = (1)^2 - 2(1) - 3 = 1 - 2 - 3 = -4 So, the vertex is at (1,4)(1, -4).

By plotting the vertex and the x-intercepts, the parabola will have its minimum point at (1,4)(1, -4), and it will pass through (1,0)(-1, 0) and (3,0)(3, 0).

Thus, the solution to the equation x22x3=0x^2 - 2x - 3 = 0 by graphing is x=3x = 3 and x=1x = -1.

Would you like a graph for better visualization or further details?

5 Questions to Explore Further:

  1. How do you determine the vertex of a quadratic equation?
  2. What is the significance of the x-intercepts in a quadratic equation?
  3. Can all quadratic equations be factored easily? If not, what methods can be used to solve them?
  4. What does the direction of a parabola (opening upwards or downwards) depend on?
  5. 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