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Solving the Equation (x-6)(x-9) = 0
This equation represents a quadratic equation in factored form. Let's break down how to solve it and understand the concept behind it.
Understanding the Zero Product Property
The Zero Product 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, (x-6) and (x-9) are the two factors. Therefore, for the product to be zero, either:
- (x - 6) = 0
- (x - 9) = 0
Solving for x
Now we solve each equation separately:
-
x - 6 = 0 Adding 6 to both sides gives us: x = 6
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x - 9 = 0 Adding 9 to both sides gives us: x = 9
Solutions
Therefore, the solutions to the equation (x-6)(x-9) = 0 are x = 6 and x = 9.
Graphical Representation
If we were to graph the function y = (x-6)(x-9), we would see that it intersects the x-axis at the points x = 6 and x = 9. These are the x-intercepts, which represent the solutions to the equation where y = 0.
Conclusion
The equation (x-6)(x-9) = 0 is a simple quadratic equation that can be solved by applying the Zero Product Property. This property helps us understand that for the product of factors to be zero, at least one of the factors must be zero. By solving for x in each factor, we find the solutions to the equation.