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Solving the Equation (x+1)(x-5) = 0
The equation (x+1)(x-5) = 0 is a quadratic equation in factored form. This form makes it incredibly easy to solve for the values of 'x' that satisfy the equation.
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.
Applying this to our equation:
- If (x+1) = 0, then the entire product will be zero.
- If (x-5) = 0, then the entire product will also be zero.
Finding the Solutions
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Solve for x in the first factor: (x + 1) = 0 x = -1
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Solve for x in the second factor: (x - 5) = 0 x = 5
Therefore, the solutions to the equation (x+1)(x-5) = 0 are x = -1 and x = 5.
Verification
We can verify these solutions by plugging them back into the original equation:
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For x = -1: (-1 + 1)(-1 - 5) = (0)(-6) = 0
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For x = 5: (5 + 1)(5 - 5) = (6)(0) = 0
Both solutions make the equation true, confirming their validity.
Conclusion
By utilizing the Zero Product Property, we easily determined the solutions for the equation (x+1)(x-5) = 0. This method proves to be a powerful tool for solving quadratic equations in factored form, simplifying the process and ensuring accurate results.