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Solving the Equation (x-1)(x-5)=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 key to solving this equation lies in the Zero Product Property. This property states that if the product of two or more factors is equal to zero, then at least one of the factors must be equal to zero.
In our case, we have two factors: (x-1) and (x-5). Therefore, for the product to be zero, one or both of these factors must be equal to zero.
Solving for x
To find the solutions, we set each factor equal to zero and solve for x:
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Factor 1: (x-1) = 0
- Add 1 to both sides: x = 1
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Factor 2: (x-5) = 0
- Add 5 to both sides: x = 5
The Solutions
Therefore, the solutions to the equation (x-1)(x-5)=0 are x = 1 and x = 5. These are the values of x that make the equation true.
Visual Representation
If we were to graph the equation y = (x-1)(x-5), we would see that the graph intersects the x-axis at the points x = 1 and x = 5. This confirms our solutions.
Conclusion
Solving equations in factored form is a straightforward process thanks to the Zero Product Property. By setting each factor equal to zero and solving, we can find the values of x that satisfy the equation.