Final answer:
The equation (x - 2)(x + 2)(x^2 + 4) = 0 has two real roots, x = 2 and x = -2, and two imaginary roots, x = 2i and x = -2i.
Step-by-step explanation:
Complex Roots, Real Roots, and Imaginary Roots of an Equation
For the equation (x - 2)(x + 2)(x^2 + 4) = 0, we can determine the complex roots as well as the possible number of real and imaginary roots. This equation is a product of three factors: two of them linear, (x - 2) and (x + 2), and one quadratic, (x^2 + 4).
The linear factors each give one real root. The solutions for (x - 2) = 0 and (x + 2) = 0 are x = 2 and x = -2, respectively.
The quadratic factor (x^2 + 4) does not have real roots because the term +4 does not allow for x^2 to be negative to cancel it out, hence there are no x-values where (x^2 + 4) = 0. Therefore, we solve the quadratic equation using the formula for complex roots where a = 1, b = 0, and c = 4:
x = (-b ± √(b^2 - 4ac)) / (2a)
Substituting in the values gives us:
x = (± √(0 - 4 * 1 * 4)) / 2
x = (± √-16) / 2
x = ± 4i / 2
x = ± 2i
Therefore, the roots of (x^2 + 4) = 0 are 2i and -2i, which are imaginary roots.
Summarizing, the equation has a total of four roots: two real roots and two imaginary roots.