Final answer:
The equation of the quadratic with a root of 1-8i is "x^2 - 2x + 65 = 0" because complex roots come in conjugate pairs and the expansion of the factors results in this equation.
Step-by-step explanation:
The correct equation of the quadratic with a root of 1-8i relies on the fact that the complex roots of polynomials with real coefficients come in conjugate pairs. Thus, if 1-8i is a root, then its conjugate 1+8i is also a root. Using these roots, we can find the quadratic equation by setting up factors (x - (1 - 8i)) and (x - (1 + 8i)) and expanding them.
The factors multiply to x² - (1 + 8i)x - (1 - 8i)x + (1 - 8i)(1 + 8i) = x² - 2x + (1 - 64) = x² - 2x + 65. So the equation of the quadratic is x² - 2x + 65 = 0.