If a quadratic equation with real coefficients has an imaginary root, then its two roots must be complex conjugates. If x + y√z is a rational root of a quadratic with rational coefficients, where x, y, and z are rational numbers and √z is irrational, then x - y√z must also be a root. The sum of the roots of a quadratic equation ax^2 + bx + c = 0 is -b/a and the product of the roots is c/a.

1. Prove that if one of the roots of a quadratic with real coefficients is imaginary, then the two roots of the quadratic are complex conjugates.

2. Given that x, y, and z are rational and the square root of z is not, show that if x+y square root of z is a root of a quadratic with rational coefficients, then x-y square root of z is also a root.

3. Prove that the sum of the roots of ax² + bx + c =0 is –b/a and that the product of the roots is c/a .