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Solving the Equation (x+8)(x-4)=0
This equation presents a simple yet crucial concept in algebra: the zero product property. This property states that if the product of two or more factors is zero, then at least one of the factors must be zero.
Let's break down how to solve the equation:
1. Identify the factors:
The equation is already factored for us: (x+8) and (x-4) are the two factors.
2. Apply the zero product property:
For the product of these two factors to be zero, at least one of them must be equal to zero. This gives us two possible scenarios:
- Scenario 1: x + 8 = 0
- Scenario 2: x - 4 = 0
3. Solve for x in each scenario:
- Scenario 1: Subtract 8 from both sides to get x = -8
- Scenario 2: Add 4 to both sides to get x = 4
Therefore, the solutions to the equation (x+8)(x-4)=0 are x = -8 and x = 4.
Understanding the Solutions:
These solutions represent the points where the graph of the equation intersects the x-axis. In other words, they are the x-intercepts of the function represented by the equation.
Generalizing the Concept:
The zero product property is a fundamental tool in solving equations that involve products of factors. It helps us efficiently find the solutions by breaking down a complex equation into simpler cases. This concept extends to equations with more than two factors as well.