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Solving Quadratic Equations: (x+a)(x+b)=0
The equation (x+a)(x+b)=0 is a simple yet fundamental form in algebra. It represents a quadratic equation in factored form and is often encountered when solving for the roots of a quadratic equation.
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 equation, the factors are (x+a) and (x+b). Therefore, for the product to be zero, either:
- (x+a) = 0 or
- (x+b) = 0
Finding the Solutions
To find the solutions, we simply solve these two linear equations:
- x + a = 0 implies x = -a
- x + b = 0 implies x = -b
Therefore, the solutions to the equation (x+a)(x+b)=0 are x = -a and x = -b.
Example
Let's consider an example:
Solve (x+2)(x-3) = 0
Following the steps above:
- x + 2 = 0 implies x = -2
- x - 3 = 0 implies x = 3
Hence, the solutions to the equation are x = -2 and x = 3.
Significance
The factored form of the quadratic equation provides a straightforward way to find its roots. This approach is often used in solving various algebraic problems, including those involving quadratic functions, graphs, and real-world applications.