x%5E2-(7%2F15)x%2B500.jpeg "(1/9000)x^2-(7/15)x+500")
Exploring the Quadratic Equation: (1/9000)x² - (7/15)x + 500
This article will delve into the quadratic equation (1/9000)x² - (7/15)x + 500, analyzing its components, exploring its properties, and discussing ways to solve it.
Understanding the Components
The equation is a quadratic equation because it is a polynomial expression of the second degree, meaning the highest power of the variable x is 2. Let's break down each term:
- (1/9000)x²: This is the quadratic term, representing the coefficient of x².
- -(7/15)x: This is the linear term, representing the coefficient of x.
- 500: This is the constant term, representing the value independent of x.
Exploring Properties
The properties of a quadratic equation are crucial for understanding its behavior and solving for its roots. Here are some key characteristics:
- Parabola: The graph of a quadratic equation is always a parabola, a U-shaped curve. The sign of the quadratic coefficient determines whether the parabola opens upwards (positive) or downwards (negative). In this case, the parabola opens upwards because the coefficient (1/9000) is positive.
- Roots: The roots of a quadratic equation are the values of x that make the equation equal to zero. These are also known as the x-intercepts, where the parabola intersects the x-axis. A quadratic equation can have two, one, or zero real roots.
- Discriminant: The discriminant of a quadratic equation, denoted by Δ, helps determine the nature of the roots. It is calculated using the formula: Δ = b² - 4ac, where a, b, and c are the coefficients of the quadratic, linear, and constant terms, respectively.
- If Δ > 0, the equation has two distinct real roots.
- If Δ = 0, the equation has one real root (a double root).
- If Δ < 0, the equation has no real roots, but two complex conjugate roots.
Solving the Equation
There are several methods to solve quadratic equations:
- Factoring: This involves rewriting the equation as a product of two linear factors. However, factoring is not always straightforward and may not be feasible for all equations.
- Quadratic Formula: This formula provides a direct solution for the roots of any quadratic equation, regardless of its complexity. The formula is: x = (-b ± √(b² - 4ac)) / 2a
- Completing the Square: This method involves manipulating the equation to create a perfect square trinomial, which can then be easily factored.
Conclusion
The quadratic equation (1/9000)x² - (7/15)x + 500 presents an intriguing example for analysis and solution. By understanding its components, properties, and solution methods, we can gain valuable insights into the behavior of quadratic equations and their applications in various fields.