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Solving the Equation (x+4)(x-7) = 0
This equation presents a simple yet important concept in algebra: 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.
Let's break down how to solve this equation:
Applying the Zero Product Property
- Identify the factors: The equation is already factored for us: (x+4) and (x-7) are the two factors.
- Set each factor to zero:
- x + 4 = 0
- x - 7 = 0
- Solve for x:
- x = -4
- x = 7
Solutions and Verification
Therefore, the solutions to the equation (x+4)(x-7) = 0 are x = -4 and x = 7.
We can verify these solutions by substituting them back into the original equation:
- For x = -4:
- (-4 + 4)(-4 - 7) = (0)(-11) = 0
- For x = 7:
- (7 + 4)(7 - 7) = (11)(0) = 0
As we can see, both solutions satisfy the equation.
Significance of the Zero Product Property
The Zero Product Property is a fundamental tool in solving polynomial equations. By factoring an equation, we can isolate the individual factors and use this property to find the solutions. This method is particularly useful for quadratic equations, where we often factor the expression into two linear factors.