um número complexo é da forma :
[tex]\text z = \text{a + b.i}[/tex]
temos :
[tex]\text z^2=2.\text i \\\\ \text{(a + b.i)}^2=2.\text i \\\\ \text a^2+(\text {b.i})^2 + 2.\text{a.b.i} = 0+2.\text i \\\\ \text a^2-\text b^2+2\text {a.b.i} = 0+2.\text i \\\\[/tex]
Parte real igual a parte real e parte imaginária igual a parte imaginária :
[tex]\underline{\text{Parte real igual a parte real}}: \\\\\ \text a^2-\text b^2 = 0 \\\\ \text a^2=\text b^2 \to \boxed{\text a = \pm \ \text b } \\\\ \underline{\text{Parte imagin{\'a}ria}}: \\\\ 2.\text{a.b.i} = 2.\text i \\\\ \text{a.b}=1 \\\\ \text{substituindo a = b}: \\\\ \text b^2 = 1 \\\\ \boxed{\text b = \pm 1} \to \boxed{\text a = \pm 1}} \\\\\underline{\text{ou seja}}: \\\\ \boxed{\text b = -1 \to \text a = -1} \\\\ \boxed{\text b = 1 \to \text a = 1}[/tex]
Portanto :
[tex]\text z = \text{a + b.i} \\\\ \huge\boxed{\text z= 1+\text i\ }\checkmark \\\\ \text{ou} \\\\ \huge\boxed{\text z = -1-\text i \ }\checkmark[/tex]
