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Solving the Equation (x-6)(x-1) = 0
The equation (x-6)(x-1) = 0 represents a quadratic equation in factored form. This form makes it very easy to solve for the values of x that satisfy the equation.
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 zero, then at least one of the factors must be zero.
In our case, we have two factors: (x-6) and (x-1). Therefore, for the product to be zero, at least one of these factors must equal zero.
Solving for x
Let's set each factor equal to zero and solve for x:
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x - 6 = 0 Adding 6 to both sides gives us: x = 6
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x - 1 = 0 Adding 1 to both sides gives us: x = 1
The Solutions
Therefore, the solutions to the equation (x-6)(x-1) = 0 are x = 6 and x = 1. These are the values of x that make the equation true.
Verification
We can verify our solutions by substituting them back into the original equation:
- For x = 6: (6 - 6)(6 - 1) = 0 * 5 = 0 (True)
- For x = 1: (1 - 6)(1 - 1) = -5 * 0 = 0 (True)
As both solutions satisfy the equation, we have successfully solved the equation (x-6)(x-1) = 0.