Question

Determinați radacinile patrate ale numărului 3 + 4i

Answer 1

$(a+bi)^2 = 3+4i \\ \\ \Rightarrow a^2+2abi-b^2 = 3+4i \\ \Rightarrow a^2-b^2+2abi = 3+4i \\ \\ \Leftrightarrow \,\,a^2-b^2 = 3\quad \wedge \quad 2ab = 4\\\\\Leftrightarrow \,\,a^2-\dfrac{4}{a^2}=3\quad \wedge\quad b = \dfrac{2}{a}\\ \\ \Rightarrow a^4-4 = 3a^2 \Rightarrow a^4-3a^2-4 = 0 \Rightarrow \\ \\ \Rightarrow (a^2+1)(a^2-4) = 0 \Rightarrow a^2 = 4 \Rightarrow a = \pm 2\\ \\ \Rightarrow b = \dfrac{2}{\pm 2}\Rightarrow b = \pm 1 \Rightarrow (a,b) = (\pm 2,\pm 1) \\ \\\\ \Rightarrow \boxed{S = \Big\{2+i,\,\,-2-i\Big\}}$