Key principle: Complex roots of polynomials with REAL coefficients always come in conjugate pairs.

If p(x) has real coefficients and alpha + ibeta is a root, then alpha - ibeta is also a root.

Quadratic ax^2 + bx + c = 0 (a, b, c real):

• If D = b^2 - 4ac < 0: roots are (-b +/- i*sqrt(|D|))/(2a), forming a conjugate pair
• Sum of roots = -b/a (real)
• Product of roots = c/a (real)
• |root|^2 = c/a (since z * z-bar = |z|^2 equals product of conjugate pair)

Forming quadratic with given complex root: If alpha + ibeta is a root, the quadratic is: x^2 - 2alpha*x + (alpha^2 + beta^2) = 0

Important: This conjugate pair property does NOT hold for polynomials with complex coefficients.