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Solving the Equation: (x+2)(x+5) = 0
This equation is a simple quadratic equation, but understanding how to solve it lays the foundation for more complex algebraic problems. Here's a breakdown of the process:
The Zero Product Property
The key to solving this equation is 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+2) and (x+5). Therefore, to make the product equal to zero, either (x+2) must be zero, or (x+5) must be zero, or both.
Solving for x
Let's consider each case separately:
Case 1: (x+2) = 0
Subtracting 2 from both sides, we get: x = -2
Case 2: (x+5) = 0
Subtracting 5 from both sides, we get: x = -5
Solution
Therefore, the solutions to the equation (x+2)(x+5) = 0 are x = -2 and x = -5.
This means that if you substitute either -2 or -5 for x in the original equation, the equation will hold true.
In Summary
- Identify the factors: The equation is already factored, giving us (x+2) and (x+5).
- Apply the Zero Product Property: For the product to be zero, at least one factor must be zero.
- Solve for x in each case: Set each factor equal to zero and solve for x.
- The solutions: The solutions are the values of x that make the equation true.
By understanding the Zero Product Property, you can efficiently solve a variety of quadratic equations, setting the stage for further exploration in algebra and beyond.