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

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

### 1. Rewrite the equation in standard form:
The given equation is already in the form:
\[
y = x^2 - 2x - 3
\]
This is the equation of a parabola.

### 2. Plot the parabola:
To graph the quadratic function, we will calculate a few values of \(x\) and the corresponding values of \(y = x^2 - 2x - 3\), 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 \(x^2 - 2x - 3 = 0\). These intercepts are the points where \(y = 0\).

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

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

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

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

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.

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

Learn how to solve the quadratic equation x^2 - 2x - 3 by graphing. This step-by-step guide covers key algebraic concepts such as factoring, finding x-intercepts, and determining the vertex of a parabola. Ideal for students in Grades 9-10.

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