(x-3).jpeg "(x-2)(x-3)")
Exploring the Expression (x - 2)(x - 3)
The expression (x - 2)(x - 3) is a simple example of a quadratic expression in factored form. Let's break down what this means and explore some of its key properties:
Understanding the Factored Form
- Factoring: When we factor a quadratic expression, we are essentially rewriting it as a product of two linear expressions. In this case, the linear expressions are (x - 2) and (x - 3).
- Roots or Zeros: The factored form tells us the roots or zeros of the quadratic. These are the values of x that make the expression equal to zero.
- Setting (x - 2) = 0, we get x = 2
- Setting (x - 3) = 0, we get x = 3
- Therefore, the roots of the expression are x = 2 and x = 3.
Expanding the Expression
To understand the relationship between the factored form and the standard form of a quadratic, let's expand the expression:
(x - 2)(x - 3) = x(x - 3) - 2(x - 3) = x² - 3x - 2x + 6 = x² - 5x + 6
This is the standard form of the quadratic expression, where the terms are arranged in descending order of their powers.
Key Points
- Roots: The roots of the quadratic (x = 2 and x = 3) represent the x-intercepts of the graph of the function y = (x - 2)(x - 3).
- Vertex: The vertex of the parabola represented by this quadratic lies at the midpoint of the roots, which is (2.5, -0.25).
- Symmetry: The parabola is symmetrical about a vertical line passing through the vertex.
Applications
The factored form of a quadratic expression is useful in many applications, including:
- Solving quadratic equations: Setting the expression equal to zero and finding the roots helps solve quadratic equations.
- Graphing quadratic functions: Knowing the roots and the vertex allows us to easily graph the function.
- Modeling real-world phenomena: Quadratic expressions are used to model various real-world scenarios, such as projectile motion, area calculations, and optimization problems.
By understanding the factored form and its properties, we can gain valuable insights into the behavior of quadratic expressions and their applications.