Given roots alpha, beta of ax^2 + bx + c = 0, find equations with roots:

Roots alpha^2, beta^2: Sum = (alpha+beta)^2 - 2alphabeta = b^2/a^2 - 2c/a. Product = (alphabeta)^2 = c^2/a^2. Equation: a^2x^2 - (b^2-2ac)x + c^2 = 0.

Roots 1/alpha, 1/beta: Replace x by 1/x in original: a/x^2 + b/x + c = 0 → cx^2 + bx + a = 0 (reverse coefficients).

Roots alpha+k, beta+k: Replace x by x-k in original: a(x-k)^2 + b(x-k) + c = 0.

Roots kalpha, kbeta: Replace x by x/k in original: a(x/k)^2 + b(x/k) + c = 0 → ax^2 + bkx + ck^2 = 0.

Roots -alpha, -beta: Replace x by -x: ax^2 - bx + c = 0 (negate middle coefficient).