(x-7)%3D0-quadratic-equation.jpeg "(x-5)(x-7)=0 Quadratic Equation")
Solving the Quadratic Equation (x-5)(x-7) = 0
This article will guide you through the process of solving the quadratic equation (x-5)(x-7) = 0.
Understanding the Equation
The equation (x-5)(x-7) = 0 is a factored quadratic equation. This means it's already in a form that makes it easy to solve.
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 zero.
Solving for x
Applying the Zero Product Property to our equation, we have two possible scenarios:
-
x - 5 = 0
Solving for x, we get x = 5. -
x - 7 = 0
Solving for x, we get x = 7.
Therefore, the solutions to the quadratic equation (x-5)(x-7) = 0 are x = 5 and x = 7.
Verification
We can verify our solutions by substituting them back into the original equation:
- For x = 5: (5 - 5)(5 - 7) = 0 * 0 = 0
- For x = 7: (7 - 5)(7 - 7) = 2 * 0 = 0
Both solutions satisfy the original equation.
Conclusion
The quadratic equation (x-5)(x-7) = 0 is a simple example of a factored quadratic equation. By utilizing the Zero Product Property, we can easily find the solutions: x = 5 and x = 7. This approach allows us to quickly solve quadratic equations without having to go through the process of expanding and then factoring the equation.