Resolva analiticamente a seguinte equação:
[tex]\sqrt{x}+\sqrt{-x}=2\quad \text{, com }\;x\in\mathbb{C}[/tex]
[tex]\sqrt{x}+\sqrt{-x}=2\quad \text{, com }\;x\in\mathbb{C}[/tex]
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√x + √(- x) = 2
(√x + √(- x))² = 2²
x + 2√(x . (- x)) - x = 4
2√(x . (- x)) = 4
2√(- x . x) = 4
2√(- x²) = 4
√(- x²) = 4/2
√(- x²) = 2
√(- x²)² = 2²
- x² = 4 . (- 1)
x² = - 4
x = ± √(- 4)
x = ± √(4 . (- 1))
x = ± √4 √(- 1)
x = ± √(2²) i
x = ± 2i
x = - 2i
x = 2i
S = {- 2i , 2i}
Verified answer
✅ Após resolver a equação irracional dada, concluímos que seu conjunto solução é:
[tex]\Large\displaystyle\text{$\begin{gathered}\boxed{\boxed{\:\:\:\bf S = \{-2i,\,2i\}\:\:\:}}\end{gathered}$}[/tex]
Seja a equação irracional:
[tex]\Large\displaystyle\text{$\begin{gathered} \sqrt{x} + \sqrt{-x} = 2,\:\:\:\textrm{com}\:x\in\mathbb{C}\end{gathered}$}[/tex]
Resolvendo esta equação, temos:
[tex]\Large\displaystyle\text{$\begin{gathered} \sqrt{x} + \sqrt{-x} = 2\end{gathered}$}[/tex]
[tex]\Large\displaystyle\text{$\begin{gathered} (\sqrt{x} + \sqrt{-x})^{2} = 2^{2}\end{gathered}$}[/tex]
[tex]\Large\displaystyle\text{$\begin{gathered}(\sqrt[\!\diagup\!\!]{x})^{\!\diagup\!\!\!\!2} + 2\cdot\sqrt{x}\cdot\sqrt{-x} + (\sqrt[\!\diagup\!\!]{-x})^{\!\diagup\!\!\!\!2} = 4\end{gathered}$}[/tex]
[tex]\Large\displaystyle\text{$\begin{gathered} x + 2\sqrt{x\cdot(-x)} + (-x) = 4\end{gathered}$}[/tex]
[tex]\Large\displaystyle\text{$\begin{gathered} x - x + 2\sqrt{-x^{2}} = 4\end{gathered}$}[/tex]
[tex]\Large\displaystyle\text{$\begin{gathered} 2\sqrt{-x^{2}} = 4\end{gathered}$}[/tex]
[tex]\Large\displaystyle\text{$\begin{gathered} \sqrt{-x^{2}} = \frac{4}{2}\end{gathered}$}[/tex]
[tex]\Large\displaystyle\text{$\begin{gathered} \sqrt{-x^{2}}= 2\end{gathered}$}[/tex]
[tex]\Large\displaystyle\text{$\begin{gathered} (\sqrt[\!\diagup\!\!]{-x^{2}})^{\!\diagup\!\!\!\!2} = 2^{2}\end{gathered}$}[/tex]
[tex]\Large\displaystyle\text{$\begin{gathered} -x^{2} = 4\end{gathered}$}[/tex]
[tex]\Large\displaystyle\text{$\begin{gathered} x^{2} = -4\end{gathered}$}[/tex]
[tex]\Large\displaystyle\text{$\begin{gathered} x = \pm\sqrt{-4}\end{gathered}$}[/tex]
[tex]\Large\displaystyle\text{$\begin{gathered} x = \pm2i\end{gathered}$}[/tex]
Portanto, o conjunto solução é:
[tex]\Large\displaystyle\text{$\begin{gathered} S = \{-2i,\,2i\}\end{gathered}$}[/tex]
[tex]\LARGE\displaystyle\text{$\begin{gathered} \underline{\boxed{\boldsymbol{\:\:\:Bons \:estudos!!\:\:\:Boa\: sorte!!\:\:\:}}}\end{gathered}$}[/tex]
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