Type 1: Biquadratic $ax^4$ + $bx^2$ + c = 0 Substitute t = $x^2$. Solve for t, then x = ±sqrt(t). Only t >= 0 gives real x.

Type 2: Reciprocal $ax^4$ + $bx^3$ + $cx^2$ + bx + a = 0 Divide by $x^2$: a($x^{2+1}$/$x^2$) + b(x+1/x) + c = 0. Substitute t = x+1/x, so $x^{2+1}$/$x^2$ = $t^{2-2}$.

Type 3: Equation with sqrt sqrt(f(x)) = g(x). Square both sides: f(x) = [g(x)]^2. But must verify g(x) >= 0 and f(x) >= 0.

Type 4: Exponential a^(2x) + b*$a^x$ + c = 0 Substitute t = $a^x$ (t > 0). Solve quadratic in t, then x = $log_a$(t).