Solving the Equation (x-4)(x+3)(-x-8) = 0
This equation represents a cubic function, which means it has a maximum of three solutions or roots. To find these roots, we can utilize the Zero Product Property:
Zero Product Property: If the product of two or more factors is zero, then at least one of the factors must be zero.
Applying this to our equation:
(x-4)(x+3)(-x-8) = 0
We have three factors:
- (x-4)
- (x+3)
- (-x-8)
For the product to equal zero, at least one of these factors must be zero. Let's solve for each case:
Case 1: (x-4) = 0
- Solving for x, we get: x = 4
Case 2: (x+3) = 0
- Solving for x, we get: x = -3
Case 3: (-x-8) = 0
- Solving for x, we get: x = -8
Therefore, the solutions to the equation (x-4)(x+3)(-x-8) = 0 are x = 4, x = -3, and x = -8.
Understanding the Solution:
These solutions represent the x-intercepts of the graph of the cubic function represented by the equation. In other words, the graph crosses the x-axis at the points (4, 0), (-3, 0), and (-8, 0).
In Summary:
By using the Zero Product Property, we successfully found the three solutions (roots) of the equation (x-4)(x+3)(-x-8) = 0, which are x = 4, x = -3, and x = -8. These solutions also represent the x-intercepts of the corresponding cubic function.