Determine, em $\mathbb{C}$, as soluções das seguintes equações
Números complexos: Infinito 12 A - Parte 3 Pág. 69 Ex. 29
Warning: Undefined variable $op in /home/acasinha/public_html/wp-content/plugins/gt-tabs/index.php on line 102
Warning: Undefined array key "GTTabs" in /home/acasinha/public_html/wp-content/plugins/gt-tabs/index.php on line 184
Enunciado
Determine as soluções das seguintes equações:
- ${x^3} + 5x = 0$
- ${x^2} + 4x + 7 = 0$
Resolução
- Ora,
$$\begin{array}{*{20}{l}}
{{x^3} + 5x = 0}& \Leftrightarrow &{x({x^2} + 5) = 0} \\
{}& \Leftrightarrow &{\begin{array}{*{20}{l}}
{x = 0}& \vee &{{x^2} + 5 = 0}
\end{array}} \\
{}& \Leftrightarrow &{\begin{array}{*{20}{l}}
{x = 0}& \vee &{{x^2} = – 5}
\end{array}} \\
{}& \Leftrightarrow &{\begin{array}{*{20}{c}}
{x = 0}& \vee &{x = – \sqrt 5 i}& \vee &{x = \sqrt 5 i}
\end{array}}
\end{array}$$ - Ora,
$$\begin{array}{*{20}{l}}
{{x^2} + 4x + 7 = 0}& \Leftrightarrow &{x = \frac{{ – 4 \pm \sqrt {16 – 28} }}{2}} \\
{}& \Leftrightarrow &{x = \frac{{ – 4 \pm 2\sqrt { – 3} }}{2}} \\
{}& \Leftrightarrow &{\begin{array}{*{20}{l}}
{x = – 2 – \sqrt 3 i}& \vee &{x = – 2 + \sqrt 3 i}
\end{array}}
\end{array}$$
