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^2$$x^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$\frac{x}{k}$^2 + b$\frac{x}{k}$ + c = 0 → $ax^2$ + bkx + $ck^2$ = 0.

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