Đáp án:
\[\left[ \begin{array}{l}
z = \frac{{1 + \sqrt 3 i}}{2}\\
z = \frac{{1 - \sqrt 3 i}}{2}
\end{array} \right.\]
Giải thích các bước giải:
Ta có:
\(\begin{array}{l}
{z^2} - z + 1 = 0\\
\Leftrightarrow {z^2} - 2.z.\frac{1}{2} + \frac{1}{4} + \frac{3}{4} = 0\\
\Leftrightarrow {\left( {z - \frac{1}{2}} \right)^2} = - \frac{3}{4}\\
\Leftrightarrow {\left( {z - \frac{1}{2}} \right)^2} = \frac{3}{4}{i^2}\\
\Leftrightarrow \left( {z - \frac{1}{2}} \right) = {\left( {\frac{{\sqrt 3 }}{2}i} \right)^2}\\
\Leftrightarrow \left[ \begin{array}{l}
z - \frac{1}{2} = \frac{{\sqrt 3 }}{2}i\\
z - \frac{1}{2} = - \frac{{\sqrt 3 }}{2}i
\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}
z = \frac{{1 + \sqrt 3 i}}{2}\\
z = \frac{{1 - \sqrt 3 i}}{2}
\end{array} \right.
\end{array}\)
