Question

Rezolvați ecuatia data, va rog​

Answer 1

$x = \dfrac{x^2+1}{x^4+1}\Rightarrow x(x^4+1) = x^2+1\\ \\ \Rightarrow x^5+x = x^2+1 \Rightarrow x^5-x^2+x-1 = 0\\ \\ \Rightarrow x^2(x^3-1)+x-1 = 0\\ \\ \Rightarrow x^2(x-1)(x^2+x+1)+(x-1) = 0\\ \\ \Rightarrow (x-1)\Big[x^2(x^2+x+1)+1\Big] = 0$

$\boxed{1}\,\,\,x-1 = 0 \Rightarrow x = 1\\\\ \boxed{2}\,\,\,x^2(x^2+x+1)+1 = 0\\ \\\text{Pentru }x^2+x+1:\\ \Delta = 1^2-4\cdot 1\cdot 1 = -3<0 \Rightarrow x^2+x+1 > 0\Big|\cdot x^2 \\ \Rightarrow x^2(x^2+x+1) \geq 0\Big|+1 \Rightarrow x^2(x^2+x+1)+1\geq 1\\ \Rightarrow x^2(x^2+x+1)+1\neq 0 \Rightarrow x\in \emptyset\\ \\\text{Din }\boxed{1}\,\cup\,\boxed{2}\Rightarrow x\in \{1\}\cup \emptyset\Rightarrow \boxed{S = \{1\}}$