(x-3).jpeg "(2x-1)(x-3)")
Exploring the Expression (2x-1)(x-3)
The expression (2x-1)(x-3) represents a product of two binomials. Let's explore its different aspects and how to work with it:
1. Expanding the Expression
The most common way to work with this expression is to expand it using the distributive property (also known as FOIL):
- First: 2x * x = 2x²
- Outer: 2x * -3 = -6x
- Inner: -1 * x = -x
- Last: -1 * -3 = 3
Combining the terms, we get: 2x² - 6x - x + 3
Simplifying further, we arrive at the expanded form: 2x² - 7x + 3
2. Finding the Roots
To find the roots of the expression, we need to solve the equation: 2x² - 7x + 3 = 0
We can use the quadratic formula to find the solutions:
x = (-b ± √(b² - 4ac)) / 2a
Where a = 2, b = -7, and c = 3.
Plugging these values into the formula, we get:
x = (7 ± √((-7)² - 4 * 2 * 3)) / (2 * 2)
x = (7 ± √25) / 4
x = (7 ± 5) / 4
Therefore, the roots of the expression are:
- x = 3
- x = 1/2
3. Graphing the Expression
The expression (2x-1)(x-3) represents a parabola when graphed. The roots we found in the previous step (x = 3 and x = 1/2) represent the x-intercepts of the parabola. The parabola opens upwards because the coefficient of the x² term is positive (2).
4. Applications
The expression (2x-1)(x-3) can be used in various applications, including:
- Modeling real-world scenarios: For example, it can represent the profit function of a company, where x represents the number of units sold.
- Solving problems in physics and engineering: The expression can be used in calculations related to motion, energy, and other physical phenomena.
5. Further Exploration
You can further explore the expression by:
- Finding the vertex of the parabola
- Determining the axis of symmetry
- Analyzing the behavior of the expression as x approaches positive or negative infinity
By understanding the different aspects of the expression (2x-1)(x-3), you gain valuable insights into its properties and potential applications.