(x-5)(x+2)(x+3)=0

2 min read Jun 17, 2024

Solving the Equation (x-5)(x+2)(x+3) = 0

This equation represents a cubic function, and finding its solutions involves understanding the Zero Product Property. This property states that if the product of multiple factors equals zero, at least one of the factors must be zero.

Understanding the Zero Product Property

In our equation, we have three factors:

(x-5)
(x+2)
(x+3)

To make the entire product equal to zero, we need to set each factor equal to zero and solve for x.

Solving for the Solutions

(x-5) = 0 Adding 5 to both sides, we get: x = 5
(x+2) = 0 Subtracting 2 from both sides, we get: x = -2
(x+3) = 0 Subtracting 3 from both sides, we get: x = -3

Therefore, the solutions to the equation (x-5)(x+2)(x+3) = 0 are x = 5, x = -2, and x = -3.

Visualizing the Solutions

These solutions represent the x-intercepts of the cubic function defined by the equation. On a graph, the function would cross the x-axis at these three points.

Conclusion

By applying the Zero Product Property, we successfully solved the equation (x-5)(x+2)(x+3) = 0 and found the three solutions: x = 5, x = -2, and x = -3. Understanding this principle is crucial for solving polynomial equations and analyzing their graphs.