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Question Number 204510 by Ghisom last updated on 20/Feb/24
solve for x≠y∧y≠z∧z≠x (exact solutions required) (√((−3+4i)x))=y (√((−3+4i)y))=z (√((−3+4i)z))=x
$$\mathrm{solve}\:\mathrm{for}\:{x}\neq{y}\wedge{y}\neq{z}\wedge{z}\neq{x} \\ $$$$\left(\mathrm{exact}\:\mathrm{solutions}\:\mathrm{required}\right) \\ $$$$\sqrt{\left(−\mathrm{3}+\mathrm{4i}\right){x}}={y} \\ $$$$\sqrt{\left(−\mathrm{3}+\mathrm{4i}\right){y}}={z} \\ $$$$\sqrt{\left(−\mathrm{3}+\mathrm{4i}\right){z}}={x} \\ $$
Answered by Frix last updated on 20/Feb/24
Obviously x=y=z=0 but this is excluded. Btw x=y=z=(−3+4i) doesn′t work anyway because (√((−3+4i)^2 ))=3−4i≠−3+4i (√((−3+4i)z))=x ⇒ z=−(((3+4i)x^2 )/(25)) (√((−3+4i)y))=−(((3+4i)x^2 )/(25)) ⇒ y=(((117−44i)x^4 )/(15625)) (√((−3+4i)x))=(((117−44i)x^4 )/(15625)) x=−(((76443−16124i)x^8 )/(6103515625)); x≠0 x^7 =−76443+16124i x^7 =5^7 e^(i(π−tan^(−1) ((16124)/(76443)))) The rest is to find the fitting solution... Sorted by the angle (circular solutions x→y→z→x) x=5e^(i(((π−tan^(−1) ((16124)/(76443)))/7))) ≈4.56727+2.03470i y=5e^(i(((3π−tan^(−1) ((16124)/(76443)))/7))) ≈1.25686+4.83945i z=5e^(−i(((3π+tan^(−1) ((16124)/(76443)))/7))) ≈.967372−4.90553i
$$\mathrm{Obviously}\:{x}={y}={z}=\mathrm{0}\:\mathrm{but}\:\mathrm{this}\:\mathrm{is}\:\mathrm{excluded}. \\ $$$$\mathrm{Btw}\:{x}={y}={z}=\left(−\mathrm{3}+\mathrm{4i}\right)\:\mathrm{doesn}'\mathrm{t}\:\mathrm{work}\:\mathrm{anyway} \\ $$$$\mathrm{because}\:\sqrt{\left(−\mathrm{3}+\mathrm{4i}\right)^{\mathrm{2}} }=\mathrm{3}−\mathrm{4i}\neq−\mathrm{3}+\mathrm{4i} \\ $$$$ \\ $$$$\sqrt{\left(−\mathrm{3}+\mathrm{4i}\right){z}}={x}\:\Rightarrow\:{z}=−\frac{\left(\mathrm{3}+\mathrm{4i}\right){x}^{\mathrm{2}} }{\mathrm{25}} \\ $$$$\sqrt{\left(−\mathrm{3}+\mathrm{4i}\right){y}}=−\frac{\left(\mathrm{3}+\mathrm{4i}\right){x}^{\mathrm{2}} }{\mathrm{25}}\:\Rightarrow\:{y}=\frac{\left(\mathrm{117}−\mathrm{44i}\right){x}^{\mathrm{4}} }{\mathrm{15625}} \\ $$$$\sqrt{\left(−\mathrm{3}+\mathrm{4i}\right){x}}=\frac{\left(\mathrm{117}−\mathrm{44i}\right){x}^{\mathrm{4}} }{\mathrm{15625}} \\ $$$${x}=−\frac{\left(\mathrm{76443}−\mathrm{16124i}\right){x}^{\mathrm{8}} }{\mathrm{6103515625}};\:{x}\neq\mathrm{0} \\ $$$${x}^{\mathrm{7}} =−\mathrm{76443}+\mathrm{16124i} \\ $$$${x}^{\mathrm{7}} =\mathrm{5}^{\mathrm{7}} \mathrm{e}^{\mathrm{i}\left(\pi−\mathrm{tan}^{−\mathrm{1}} \:\frac{\mathrm{16124}}{\mathrm{76443}}\right)} \\ $$$$\mathrm{The}\:\mathrm{rest}\:\mathrm{is}\:\mathrm{to}\:\mathrm{find}\:\mathrm{the}\:\mathrm{fitting}\:\mathrm{solution}… \\ $$$$\mathrm{Sorted}\:\mathrm{by}\:\mathrm{the}\:\mathrm{angle}\:\left(\mathrm{circular}\:\mathrm{solutions}\right. \\ $$$$\left.{x}\rightarrow{y}\rightarrow{z}\rightarrow{x}\right) \\ $$$${x}=\mathrm{5e}^{\mathrm{i}\left(\frac{\pi−\mathrm{tan}^{−\mathrm{1}} \:\frac{\mathrm{16124}}{\mathrm{76443}}}{\mathrm{7}}\right)} \approx\mathrm{4}.\mathrm{56727}+\mathrm{2}.\mathrm{03470i} \\ $$$${y}=\mathrm{5e}^{\mathrm{i}\left(\frac{\mathrm{3}\pi−\mathrm{tan}^{−\mathrm{1}} \:\frac{\mathrm{16124}}{\mathrm{76443}}}{\mathrm{7}}\right)} \approx\mathrm{1}.\mathrm{25686}+\mathrm{4}.\mathrm{83945i} \\ $$$${z}=\mathrm{5e}^{−\mathrm{i}\left(\frac{\mathrm{3}\pi+\mathrm{tan}^{−\mathrm{1}} \:\frac{\mathrm{16124}}{\mathrm{76443}}}{\mathrm{7}}\right)} \approx.\mathrm{967372}−\mathrm{4}.\mathrm{90553i} \\ $$
Commented by Ghisom last updated on 20/Feb/24
thank you

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