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Solving the Equation (x - 8)(x + 4) = 0
This equation presents a simple yet fundamental concept in algebra: the zero product property. This property states that if the product of two or more factors equals zero, then at least one of the factors must be zero.
Let's break down the steps to solve this equation:
1. Understand the Zero Product Property
The equation (x - 8)(x + 4) = 0 implies that either (x - 8) = 0 or (x + 4) = 0.
2. Solve for x in each factor
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For (x - 8) = 0:
- Add 8 to both sides of the equation:
- x - 8 + 8 = 0 + 8
- x = 8
- Add 8 to both sides of the equation:
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For (x + 4) = 0:
- Subtract 4 from both sides of the equation:
- x + 4 - 4 = 0 - 4
- x = -4
- Subtract 4 from both sides of the equation:
3. The Solutions
Therefore, the solutions to the equation (x - 8)(x + 4) = 0 are x = 8 and x = -4.
4. Verification
We can verify our solutions by plugging them back into the original equation:
- For x = 8:
- (8 - 8)(8 + 4) = 0 * 12 = 0
- For x = -4:
- (-4 - 8)(-4 + 4) = -12 * 0 = 0
Since both solutions result in 0, we have confirmed that they are indeed the correct solutions.
In Conclusion:
The equation (x - 8)(x + 4) = 0 is solved by utilizing the zero product property, leading to two solutions: x = 8 and x = -4. This simple equation demonstrates a crucial concept in algebra and serves as a foundation for understanding more complex equations.