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Solving the Equation (x-6)(x-5) = 0
This equation represents a simple quadratic equation in factored form. To solve for the values of x, we can use the Zero Product Property:
Zero Product Property: If the product of two or more factors is zero, then at least one of the factors must be zero.
Following this property, we can set each factor in the equation equal to zero:
- x - 6 = 0
- x - 5 = 0
Now, solve for x in each equation:
- x = 6
- x = 5
Therefore, the solutions to the equation (x-6)(x-5) = 0 are x = 6 and x = 5.
Understanding the Solution
These solutions represent the x-intercepts of the quadratic function represented by the equation. In other words, the graph of the function would cross the x-axis at the points (6, 0) and (5, 0).
Visualizing the Equation
Imagine the equation as a parabola opening upwards. The x-intercepts are the points where the parabola intersects the x-axis. The solutions we found (x = 6 and x = 5) represent these points.
Conclusion
By applying the Zero Product Property, we efficiently solved the equation (x-6)(x-5) = 0 and found the solutions x = 6 and x = 5. This method is essential for understanding the relationship between factored quadratic equations and their corresponding solutions.