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Solving the Equation (k+1)(k-5)=0
This equation represents a quadratic equation in factored form. Let's break down how to solve it and understand what it means.
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 case, we have two factors: (k+1) and (k-5). Therefore, for the entire equation to equal zero, one or both of these factors must be equal to zero.
Finding the Solutions
Let's set each factor equal to zero and solve:
- Factor 1: k + 1 = 0
- Subtract 1 from both sides: k = -1
- Factor 2: k - 5 = 0
- Add 5 to both sides: k = 5
Solutions and Interpretation
Therefore, the solutions to the equation (k+1)(k-5)=0 are k = -1 and k = 5. These solutions represent the values of k that make the equation true.
This equation can be interpreted graphically as well. The equation represents a parabola that intersects the x-axis at the points (-1,0) and (5,0). These points of intersection correspond to the solutions we found.
In summary:
- The Zero Product Property is used to solve factored quadratic equations.
- The solutions to the equation (k+1)(k-5)=0 are k = -1 and k = 5.
- These solutions represent the x-intercepts of the parabola represented by the equation.