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Solving the Equation (x+3)(x-5)=0
This equation presents a simple quadratic equation in factored form, which makes it very easy to solve for the values of x.
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+3) and (x-5). Their product equals zero, so at least one of them must be zero.
Solving for x
Therefore, we can set each factor equal to zero and solve for x:
-
x + 3 = 0
Subtracting 3 from both sides gives us: x = -3 -
x - 5 = 0 Adding 5 to both sides gives us: x = 5
The Solutions
We have found two solutions to the equation:
- x = -3
- x = 5
These are the values of x that make the original equation true.
Verification
We can verify our solutions by plugging them back into the original equation:
- For x = -3: (-3 + 3)(-3 - 5) = (0)(-8) = 0
- For x = 5: (5 + 3)(5 - 5) = (8)(0) = 0
Both solutions satisfy the equation, confirming their validity.
Conclusion
By utilizing the Zero Product Property, we easily solved the equation (x+3)(x-5)=0, finding two distinct solutions: x = -3 and x = 5.