We denote by (E) the equation $$z ^ { 4 } + 4 z ^ { 2 } + 16 = 0$$ of unknown complex number $z$.
Solve in $\mathbb { C }$ the equation $Z ^ { 2 } + 4 Z + 16 = 0$. Write the solutions of this equation in exponential form.
We denote by $a$ the complex number whose modulus is equal to 2 and one of whose arguments is equal to $\frac { \pi } { 3 }$. Calculate $a ^ { 2 }$ in algebraic form. Deduce the solutions in $\mathbb { C }$ of the equation $z ^ { 2 } = - 2 + 2 \mathrm { i } \sqrt { 3 }$. Write the solutions in algebraic form.
Organized presentation of knowledge We assume it is known that for every complex number $z = x + \mathrm { i } y$ where $x \in \mathbb { R }$ and $y \in \mathbb { R }$, the conjugate of $z$ is the complex number $\bar{z}$ defined by $\bar{z} = x - \mathrm { i } y$. Prove that:
For all complex numbers $z _ { 1 }$ and $z _ { 2 } , \overline { z _ { 1 } z _ { 2 } } = \overline { z _ { 1 } } \cdot \overline { z _ { 2 } }$.
For every complex number $z$ and every non-zero natural integer $n , \overline { z ^ { n } } = ( \bar { z } ) ^ { n }$.
Prove that if $z$ is a solution of equation (E) then its conjugate $\bar { z }$ is also a solution of (E). Deduce the solutions in $\mathbb { C }$ of equation (E). We will assume that (E) has at most four solutions.
We denote by (E) the equation
$$z ^ { 4 } + 4 z ^ { 2 } + 16 = 0$$
of unknown complex number $z$.
\begin{enumerate}
\item Solve in $\mathbb { C }$ the equation $Z ^ { 2 } + 4 Z + 16 = 0$.\\
Write the solutions of this equation in exponential form.\\
\item We denote by $a$ the complex number whose modulus is equal to 2 and one of whose arguments is equal to $\frac { \pi } { 3 }$.\\
Calculate $a ^ { 2 }$ in algebraic form.\\
Deduce the solutions in $\mathbb { C }$ of the equation $z ^ { 2 } = - 2 + 2 \mathrm { i } \sqrt { 3 }$. Write the solutions in algebraic form.\\
\item Organized presentation of knowledge\\
We assume it is known that for every complex number $z = x + \mathrm { i } y$ where $x \in \mathbb { R }$ and $y \in \mathbb { R }$, the conjugate of $z$ is the complex number $\bar{z}$ defined by $\bar{z} = x - \mathrm { i } y$.\\
Prove that:
\begin{itemize}
\item For all complex numbers $z _ { 1 }$ and $z _ { 2 } , \overline { z _ { 1 } z _ { 2 } } = \overline { z _ { 1 } } \cdot \overline { z _ { 2 } }$.
\item For every complex number $z$ and every non-zero natural integer $n , \overline { z ^ { n } } = ( \bar { z } ) ^ { n }$.
\end{itemize}
\item Prove that if $z$ is a solution of equation (E) then its conjugate $\bar { z }$ is also a solution of (E).\\
Deduce the solutions in $\mathbb { C }$ of equation (E). We will assume that (E) has at most four solutions.
\end{enumerate}