(x-b)-0.jpeg "(x-a)(x-b) 0")
Solving (x-a)(x-b) = 0
The equation (x-a)(x-b) = 0 is a fundamental concept in algebra that arises in various contexts, particularly when dealing with quadratic equations and finding their roots. This article will explore how to solve this equation and understand its implications.
The Zero Product Property
The key to solving this equation lies in the Zero Product Property. This property states that if the product of two or more factors is equal to zero, then at least one of the factors must be zero.
Applying this to our equation, we have two factors: (x-a) and (x-b). For their product to be zero, one or both of these factors must be zero:
- x - a = 0
- x - b = 0
Finding the Solutions
To find the solutions for x, we simply solve these two equations:
- x = a
- x = b
These are the two roots or solutions to the equation (x-a)(x-b) = 0.
Interpretation
The roots of the equation represent the x-values where the expression (x-a)(x-b) equals zero. Graphically, these roots correspond to the points where the graph of the function y = (x-a)(x-b) intersects the x-axis.
Example
Let's take the equation (x-3)(x+2) = 0. Following the steps above, we get:
- x - 3 = 0 => x = 3
- x + 2 = 0 => x = -2
Therefore, the solutions to the equation (x-3)(x+2) = 0 are x = 3 and x = -2.
Conclusion
The equation (x-a)(x-b) = 0 is a simple yet crucial equation in algebra. By understanding the Zero Product Property and applying it, we can easily find the roots of this equation. These roots have significant implications in various mathematical contexts, including graphing functions and solving real-world problems.