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Solving the Equation (x-a)(x+b) = 0
The equation (x-a)(x+b) = 0 is a simple quadratic equation in factored form. This form allows us to easily find its solutions, which are the values of x that satisfy the equation.
Understanding the Principle
The equation states that the product of two factors, (x-a) and (x+b), equals zero. 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.
Finding the Solutions
Using the Zero Product Property, we can set each factor equal to zero and solve for x:
- x - a = 0
- Solving for x, we get x = a
- x + b = 0
- Solving for x, we get x = -b
Therefore, the solutions to the equation (x-a)(x+b) = 0 are x = a and x = -b.
Example
Let's solve the equation (x - 3)(x + 5) = 0.
Following the steps above:
- x - 3 = 0
- Solving for x, we get x = 3
- x + 5 = 0
- Solving for x, we get x = -5
Therefore, the solutions to the equation (x - 3)(x + 5) = 0 are x = 3 and x = -5.
Conclusion
The equation (x-a)(x+b) = 0 is a simple quadratic equation that can be easily solved by applying the Zero Product Property. This principle allows us to find the solutions by setting each factor equal to zero and solving for x. This approach provides a straightforward method for finding the values of x that satisfy the equation.