Question

Resolva analiticamente a seguinte equação:

$\sqrt{x}+\sqrt{-x}=2\quad \text{, com }\;x\in\mathbb{C}$

Answer 1

√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}

Answer 2

✅ Após resolver a equação irracional dada, concluímos que seu conjunto solução é:

                        $\Large\displaystyle\text{ \begin{gathered}\boxed{\boxed{\:\:\:\bf S = \{-2i,\,2i\}\:\:\:}}\end{gathered} }$          

 

Seja a equação irracional:

               $\Large\displaystyle\begin{gathered} \sqrt{x} + \sqrt{-x} = 2,\:\:\:\textrm{com}\:x\in\mathbb{C}\end{gathered}$

Resolvendo esta equação, temos:

                                              $\Large\displaystyle\begin{gathered} \sqrt{x} + \sqrt{-x} = 2\end{gathered}$

                                        $\Large\displaystyle\begin{gathered} (\sqrt{x} + \sqrt{-x})^{2} = 2^{2}\end{gathered}$

$\Large\displaystyle\begin{gathered}(\sqrt[\!\diagup\!\!]{x})^{\!\diagup\!\!\!\!2} + 2\cdot\sqrt{x}\cdot\sqrt{-x} + (\sqrt[\!\diagup\!\!]{-x})^{\!\diagup\!\!\!\!2} = 4\end{gathered}$

                    $\Large\displaystyle\begin{gathered} x + 2\sqrt{x\cdot(-x)} + (-x) = 4\end{gathered}$

                                    $\Large\displaystyle\text{ \begin{gathered} x - x + 2\sqrt{-x^{2}} = 4\end{gathered} }$

                                                      $\Large\displaystyle\text{ \begin{gathered} 2\sqrt{-x^{2}} = 4\end{gathered} }$

                                                        $\Large\displaystyle\text{ \begin{gathered} \sqrt{-x^{2}} = \frac{4}{2}\end{gathered} }$

                                                        $\Large\displaystyle\text{ \begin{gathered} \sqrt{-x^{2}}= 2\end{gathered} }$

                                                   $\Large\displaystyle\text{ \begin{gathered} (\sqrt[\!\diagup\!\!]{-x^{2}})^{\!\diagup\!\!\!\!2} = 2^{2}\end{gathered} }$

                                                              $\Large\displaystyle\begin{gathered} -x^{2} = 4\end{gathered}$

                                                                 $\Large\displaystyle\begin{gathered} x^{2} = -4\end{gathered}$

                                                                   $\Large\displaystyle\begin{gathered} x = \pm\sqrt{-4}\end{gathered}$

                                                                   $\Large\displaystyle\begin{gathered} x = \pm2i\end{gathered}$

Portanto, o conjunto solução é:

                    $\Large\displaystyle\begin{gathered} S = \{-2i,\,2i\}\end{gathered}$

$\LARGE\displaystyle\text{ \begin{gathered} \underline{\boxed{\boldsymbol{\:\:\:Bons \:estudos!!\:\:\:Boa\: sorte!!\:\:\:}}}\end{gathered} }$

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