Đáp án:
\[\left| w \right| = \sqrt {457} \]
Giải thích các bước giải:
Ta có:
\(\begin{array}{l}
z = x + yi\\
\left| z \right| - 2\overline z = - 7 + 3i + z\\
\Leftrightarrow \sqrt {{x^2} + {y^2}} - 2\left( {x - yi} \right) = - 7 + 3i + x + yi\\
\Leftrightarrow \sqrt {{x^2} + {y^2}} - 2x + 2yi = - 7 + 3i + x + yi\\
\Leftrightarrow \left( {\sqrt {{x^2} + {y^2}} - 3x + 7} \right) + \left( {y - 3} \right)i = 0\\
\Leftrightarrow \left\{ \begin{array}{l}
\sqrt {{x^2} + {y^2}} - 3x + 7 = 0\\
y - 3 = 0
\end{array} \right.\\
\Leftrightarrow \left\{ \begin{array}{l}
y = 3\\
\sqrt {{x^2} + 9} = 3x - 7
\end{array} \right.\\
\Leftrightarrow \left\{ \begin{array}{l}
y = 3\\
x \ge \frac{7}{3}\\
{x^2} + 9 = 9{x^2} - 42x + 49
\end{array} \right.\\
\Leftrightarrow \left\{ \begin{array}{l}
y = 3\\
x \ge \frac{7}{3}\\
8{x^2} - 42x + 40 = 0
\end{array} \right.\\
\Leftrightarrow \left\{ \begin{array}{l}
y = 3\\
x \ge \frac{7}{3}\\
\left[ \begin{array}{l}
x = 4\\
x = \frac{5}{4}
\end{array} \right.
\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}
x = 4\\
y = 3
\end{array} \right. \Rightarrow z = 4 + 3i\\
w = 1 - z + {z^2} = 1 - \left( {4 + 3i} \right) + {\left( {4 + 3i} \right)^2} = 1 - 4 - 3i + 16 + 24i + 9{i^2} = 4 + 21i\\
\Rightarrow \left| w \right| = \sqrt {{4^2} + {{21}^2}} = \sqrt {457}
\end{array}\)
Vậy \(\left| w \right| = \sqrt {457} \)
