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Solving the Equation (x-6)(x-6) = 0
This equation represents a simple quadratic equation in factored form. Let's break down how to solve 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) is a factor that appears twice.
Solving for x
To find the solutions, we apply the Zero Product Property:
- Set each factor equal to zero:
- (x-6) = 0
- Solve for x:
- x = 6
The Solution
Therefore, the solution to the equation (x-6)(x-6) = 0 is x = 6. This solution has a multiplicity of 2, meaning it appears twice in the factored form of the equation.
Visualizing the Solution
Graphically, this equation represents a parabola that intersects the x-axis at the point x = 6. The fact that the solution has a multiplicity of 2 indicates that the parabola touches the x-axis at this point, but doesn't cross it.
Conclusion
Solving the equation (x-6)(x-6) = 0 involves utilizing the Zero Product Property, which leads to a single solution, x = 6, with a multiplicity of 2. Understanding the Zero Product Property is essential for solving factored quadratic equations.