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Solving the Equation (x-8)(x-3) = 0
This equation represents a quadratic equation in factored form. Let's break down how to solve it.
Understanding 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 zero, then at least one of the factors must be zero.
In our equation, we have two factors: (x-8) and (x-3). Therefore, for the product to be zero, at least one of these factors must equal zero.
Solving for x
Let's set each factor equal to zero and solve for x:
Factor 1: (x-8) = 0
- Add 8 to both sides: x = 8
Factor 2: (x-3) = 0
- Add 3 to both sides: x = 3
Solutions
Therefore, the solutions to the equation (x-8)(x-3) = 0 are x = 8 and x = 3.
Verification
We can verify our solutions by substituting them back into the original equation:
- For x = 8: (8-8)(8-3) = 0 * 5 = 0
- For x = 3: (3-8)(3-3) = -5 * 0 = 0
Since both solutions result in 0, we have verified that they are correct.
Conclusion
In summary, to solve an equation in factored form, we utilize the Zero Product Property. We set each factor equal to zero and solve for the variable. In this case, the solutions for the equation (x-8)(x-3) = 0 are x = 8 and x = 3.