(x^2-4)(x^2+6x+9)=

2 min read Jun 17, 2024

Factoring and Solving the Equation: (x^2-4)(x^2+6x+9) = 0

This problem involves factoring a quadratic equation and finding its roots. Let's break it down step-by-step:

1. Factoring the Equation

The equation (x^2-4)(x^2+6x+9) = 0 is already in a factored form. However, we can further simplify it by recognizing the patterns:

(x^2-4) is a difference of squares, which factors into (x+2)(x-2).
(x^2+6x+9) is a perfect square trinomial, which factors into (x+3)(x+3) or (x+3)^2.

Therefore, the fully factored equation becomes:

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

2. Finding the Roots

To find the roots of the equation (where it equals zero), we set each factor equal to zero and solve for x:

x+2 = 0 => x = -2
x-2 = 0 => x = 2
(x+3)^2 = 0 => x = -3 (this root has a multiplicity of 2)

3. Solution

The solutions to the equation (x^2-4)(x^2+6x+9) = 0 are:

x = -2
x = 2
x = -3 (with multiplicity 2)

This means the equation has three distinct roots: -2, 2, and -3. The root -3 appears twice, indicating a double root.

Conclusion

By factoring the given equation and solving for x, we determined the roots of the equation. This understanding helps us analyze the behavior of the function represented by the equation and its graph.