Question

what quadratic equations has roots of 3+i and 3-i

Answer 1

If the roots are 3 + i and 3 − i, the monic quadratic is
(x − (3 + i))(x − (3 − i)) = (x − 3 − i)(x − 3 + i) = (x − 3)^2 + 1 = x^2 − 6x + 10.

Any nonzero constant multiple a(x^2 − 6x + 10) = 0 (a ≠ 0) gives the same roots.