Cho hai số phức z, w thỏa mãn $\left| z+2w \right|=3$, $\left|...

Ta có $\begin{aligned}
& \left| z+2w \right|=3\Leftrightarrow {{\left| z+2w \right|}^{2}}=9\Leftrightarrow \left( z+2w \right).\left( \overline{z+2w} \right)=9 \\
& \Leftrightarrow \left( z+2w \right).\left( \overline{z}+2\overline{w} \right)=9\Leftrightarrow z.\overline{z}+2\left( z.\overline{w}+\overline{z}.w \right)+4w.\overline{z}=9\Leftrightarrow {{\left| z \right|}^{2}}+2P+4{{\left| w \right|}^{2}}=9\ \ \ \left( 1 \right) \\
\end{aligned}$
Tương tự:
$\begin{aligned}
& \left| 2z+3w \right|=6\Leftrightarrow \left( 2z+3w \right).\left( 2\overline{z}+3\overline{w} \right)=36\Leftrightarrow 4{{\left| z \right|}^{2}}+6P+9{{\left| w \right|}^{2}}=36\ \ \ \left( 2 \right) \\
& \left| z+4w \right|=7\Leftrightarrow \left( z+4w \right).\left( \overline{z}+4\overline{w} \right)=49\Leftrightarrow {{\left| z \right|}^{2}}+4P+16{{\left| w \right|}^{2}}=49\ \ \ \left( 3 \right) \\
\end{aligned}$
Từ (1), (2), (3) ta có $\left\{ \begin{aligned}
& {{\left| z \right|}^{2}}=33 \\
& P=-28 \\
& {{\left| w \right|}^{2}}=8 \\
\end{aligned} \right.\Rightarrow P=-28$.