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Solving the Equation (x-3)(x-5) = 0
This equation represents a simple quadratic equation that can be solved using 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 the steps to solve this equation:
1. Identify the factors
The equation (x-3)(x-5) = 0 already presents us with two factors: (x-3) and (x-5).
2. Apply the Zero Product Property
Since the product of these factors is zero, we know that at least one of them must equal zero. Therefore, we can set each factor equal to zero and solve for x:
- x - 3 = 0
- x - 5 = 0
3. Solve for x
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x - 3 = 0 Adding 3 to both sides: x = 3
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x - 5 = 0 Adding 5 to both sides: x = 5
Solution
Therefore, the solutions to the equation (x-3)(x-5) = 0 are x = 3 and x = 5.
Understanding the Solution
These solutions represent the x-intercepts of the quadratic function represented by the equation. In other words, these are the points where the graph of the function crosses the x-axis.
It's important to remember that the Zero Product Property is a powerful tool for solving equations where factors are multiplied together and set equal to zero.