數與座標系/6 複數與複數平面-例

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$\begin{align} & \left\{ \begin{align} & f\left( x \right)=\left( 1+i \right)x^{2}-2\left( \gamma +i \right)x+5-3i,\gamma \in R \\ & \exists \alpha \in R,f\left( \alpha \right)=0 \\ \end{align} \right. \\ & \gamma =?\alpha =?\beta =? \\ & f\left( \alpha \right)=0\Rightarrow \left( 1+i \right)\alpha ^{2}-2\left( \gamma +i \right)\alpha +5-3i=0 \\ & \Rightarrow \alpha ^{2}+\alpha ^{2}i-2\gamma \alpha -2\alpha i+5-3i=0 \\ & \Rightarrow \alpha ^{2}-2\gamma \alpha +5+\left( \alpha ^{2}-2\alpha -3 \right)i=0 \\ & \Rightarrow \left\{ \begin{align} & \alpha ^{2}-2\gamma \alpha +5=0 \\ & \alpha ^{2}-2\alpha -3=0 \\ \end{align} \right. \\ & \alpha ^{2}-2\alpha -3=0\Rightarrow \left( \alpha -3 \right)\cdot \left( \alpha +1 \right)=0\Rightarrow \left\{ \begin{align} & \alpha =3\# \\ & \alpha =-1\# \\ \end{align} \right. \\ & \left\{ \begin{align} & \alpha ^{2}-2\gamma \alpha +5=0 \\ & \alpha =3 \\ \end{align} \right.\Rightarrow 9-6\gamma +5=0\Rightarrow \gamma =\frac{7}{3}\# \\ & \left\{ \begin{align} & \alpha \beta =\frac{5-3i}{1+i} \\ & \alpha =3 \\ \end{align} \right.\Rightarrow \beta =\frac{1}{3}\times \frac{5-3i}{1+i}=\frac{1}{3}\times \frac{5-3i}{1+i}\times \frac{1-i}{1-i}=\frac{1}{3}\times \frac{5-3-8i}{2}=\frac{1-4i}{3}\# \\ & {\color{Red}\alpha =3,\beta =\frac{1-4i}{3},\gamma =\frac{7}{3}\# }\\ & \left\{ \begin{align} & \alpha ^{2}-2\gamma \alpha +5=0 \\ & \alpha =-1 \\ \end{align} \right.\Rightarrow 1+2\gamma +5=0\Rightarrow \gamma =-3\# \\ & \left\{ \begin{align} & \alpha \beta =\frac{5-3i}{1+i} \\ & \alpha =-1 \\ \end{align} \right.\Rightarrow \beta =-\frac{5-3i}{1+i}=-\frac{5-3i}{1+i}\times \frac{1-i}{1-i}=-\frac{5-3-8i}{2}=-1+4i\# \\ & {\color{Red}\alpha =-1,\beta =-1+4i,\gamma =-3\#} \\ & \\ \end{align}$

取自"http://www.come2pass.com/wiki/index.php/%E6%95%B8%E8%88%87%E5%BA%A7%E6%A8%99%E7%B3%BB/6_%E8%A4%87%E6%95%B8%E8%88%87%E8%A4%87%E6%95%B8%E5%B9%B3%E9%9D%A2-%E4%BE%8B"

數與座標系/6 複數與複數平面-例

出自高材生

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8

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