Final answer:
The quadratic equation with roots 3+9i and 3-9i is x² - 6x + 90 = 0, since the sum and product of the roots translate to the coefficients b and c in the equation ax² + bx + c = 0, leading to b = -6 and c = 90.
Step-by-step explanation:
To find which quadratic equation has the roots 3+9i and 3-9i, we can use the fact that the sum and product of the roots are related to the coefficients of a quadratic equation in the form ax² + bx + c = 0, where 'a' is the coefficient of the x² term, 'b' is the coefficient of the x term, and 'c' is the constant term.
The sum of the roots (α + β) is equal to -b/a, and the product of the roots (α⋅β) is equal to c/a. For the roots 3+9i and 3-9i, the sum is (3+9i) + (3-9i) = 6 and the product is (3+9i)⋅(3-9i) = 9 + 81 = 90.
Therefore, the quadratic equation with these roots will have 'a' equal to 1 (since the leading term is x² with an implied coefficient of 1), 'b' equal to -6 (from the negative of the sum of the roots), and 'c' equal to 90 (the product of the roots). The correct quadratic equation is x² - 6x + 90 = 0, which corresponds to option 1).