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Question Number 72421 by TawaTawa last updated on 28/Oct/19

Answered by mind is power last updated on 28/Oct/19

$$\mathrm{let}\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d}\mathrm{root}[\mathrm{in}\mathbb{C}$$$$\mathrm{a}+\mathrm{b}+\mathrm{c}+\mathrm{d}=-2$$$$\mathrm{a}+\mathrm{b}=2$$$$\mathrm{c}+\mathrm{d}=-4$$$$\mathrm{ab}+\mathrm{ac}+\mathrm{ad}+\mathrm{bc}+\mathrm{bd}+\mathrm{cd}=2..\mathrm{s}$$$$(\mathrm{a}+\mathrm{b})(\mathrm{c}+\mathrm{d})=\mathrm{ac}+\mathrm{ad}+\mathrm{bc}+\mathrm{bd}=2.-4=-8$$$$\Rightarrow \mathrm{s}\iff \mathrm{ab}+\mathrm{cd}=10$$$$\mathrm{abcd}=25=\mathrm{ab}.\mathrm{cd}$$$$\mathrm{ab},\mathrm{cd}\mathrm{root}\mathrm{of}$$$${\mathrm{X}}^2-10\mathrm{X}+25=0$$$$\mathrm{ab}=\mathrm{cd}=5,\mathrm{a}+\mathrm{b}=2,\mathrm{c}+\mathrm{d}=-4$$$$\mathrm{a},\mathrm{b}\mathrm{root}\mathrm{of}{\mathrm{x}}^2-2\mathrm{x}+5=0$$$$\Rightarrow (\mathrm{a},\mathrm{b})\in \{(1-2\mathrm{i},1+2\mathrm{i}),(1+2\mathrm{i},1-2\mathrm{i})\}$$$$(\mathrm{c},\mathrm{d})\mathrm{roots}\mathrm{of}{\mathrm{x}}^2+4\mathrm{x}+5=0$$$$(\mathrm{c},\mathrm{d})\in \{-2+\mathrm{i},-2-\mathrm{i}\}$$$$\mathrm{roots}\mathrm{are}1\underset{-}{+}2\mathrm{i},-2\underset{-}{+}\mathrm{i}$$$$$$

Commented by TawaTawa last updated on 28/Oct/19

$$\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir}$$

Commented by mind is power last updated on 28/Oct/19

$$\mathrm{y},\mathrm{re}\mathrm{welcom}$$

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