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Solving the Equation (x - 3)(x - 2) = 0
This equation represents a simple quadratic equation in factored form. Let's explore how to solve it:
Understanding the Zero Product Property
The equation relies on the Zero Product Property, which states:
If the product of two or more factors is zero, then at least one of the factors must be zero.
In our case, the factors are (x - 3) and (x - 2). To make the product equal to zero, one or both of these factors must equal zero.
Solving for x
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Factor 1: (x - 3) = 0
- Add 3 to both sides: x = 3
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Factor 2: (x - 2) = 0
- Add 2 to both sides: x = 2
Therefore, the solutions to the equation (x - 3)(x - 2) = 0 are x = 3 and x = 2.
Interpretation
These solutions represent the x-intercepts of the quadratic function represented by the equation. In other words, the graph of the function crosses the x-axis at the points x = 3 and x = 2.
Conclusion
By understanding the Zero Product Property, we can easily solve factored quadratic equations like (x - 3)(x - 2) = 0. This property allows us to find the values of x that make the equation true, which are the roots or solutions of the equation.