Complex Solve 1

$\begin{array}{l}{{z}^{4}}-2{{z}^{2}}+4=0,z\in C\\\text{Let }w={{z}^{2}}\Rightarrow \\{{w}^{2}}-2w+4=0\\\left( {{{w}^{2}}-2w+1} \right)+4-1=0\\{{\left( {w-1} \right)}^{2}}-3=0\\{{\left( {w-1} \right)}^{2}}-\left( {i\sqrt{3}} \right)=0\\\left( {w-1-i\sqrt{3}} \right)\left( {w-1+i\sqrt{3}} \right)=0\\w-1-i\sqrt{3}=0\Rightarrow \\{{z}^{2}}=1+i\sqrt{3}=2cis\left( {\frac{{\pi \left( {6n+1} \right)}}{3}} \right)\\z=\sqrt{2}cis\left( {\frac{{\pi \left( {6n+1} \right)}}{6}} \right)\\n=0,{{z}_{1}}=\sqrt{2}cis\left( {\frac{{\pi \left( 1 \right)}}{6}} \right)=\sqrt{2}\left( {\frac{{\sqrt{3}}}{2}+i\frac{1}{2}} \right)\\n=1,{{z}_{2}}=\sqrt{2}cis\left( {\frac{{\pi \left( 7 \right)}}{6}} \right)=\sqrt{2}\left( {-\frac{{\sqrt{3}}}{2}-i\frac{1}{2}} \right)\\w-1+i\sqrt{3}=0\Rightarrow \\{{z}^{2}}=1-i\sqrt{3}=2cis\left( {\frac{{\pi \left( {6n-1} \right)}}{3}} \right)\\z=\sqrt{2}cis\left( {\frac{{\pi \left( {6n-1} \right)}}{6}} \right)\\n=0,{{z}_{1}}=\sqrt{2}cis\left( {\frac{{\pi \left( {-1} \right)}}{6}} \right)=\sqrt{2}\left( {\frac{{\sqrt{3}}}{2}-i\frac{1}{2}} \right)\\n=1,{{z}_{2}}=\sqrt{2}cis\left( {\frac{{\pi \left( 5 \right)}}{6}} \right)=\sqrt{2}\left( {-\frac{{\sqrt{3}}}{2}+i\frac{1}{2}} \right)\end{array}$

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Complex Solve 1

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