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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

let a,b,c,d root[in C a+b+c+d=−2 a+b=2 c+d=−4 ab+ac+ad+bc+bd+cd=2..s (a+b)(c+d)=ac+ad+bc+bd=2.−4=−8 ⇒s⇔ab+cd=10 abcd=25=ab.cd ab,cd root of X^2 −10X+25=0 ab=cd=5,a+b=2,c+d=−4 a,b root of x^2 −2x+5=0 ⇒(a,b)∈{(1−2i,1+2i),(1+2i,1−2i)} (c,d)roots of x^2 +4x+5=0 (c,d)∈{−2+i,−2−i} roots are 1+_− 2i,−2+_− i

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

Commented by TawaTawa last updated on 28/Oct/19

God bless you sir

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

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

y,re welcom

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

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