Question Number 68141 by MJS last updated on 06/Sep/19 $$\mathrm{the}\:\mathrm{2}\:\mathrm{formulas}\:\mathrm{for}\:\mathrm{solving}\:\int\frac{{dx}}{{x}^{\mathrm{3}} +{px}+{q}}\:\mathrm{with} \\ $$$$“\mathrm{nasty}''\:\mathrm{solutions}\:\mathrm{of}\:{x}^{\mathrm{3}} +{px}+{q}=\mathrm{0}\:\mathrm{with}\:{p},\:{q}\:\in\mathbb{R} \\ $$$$ \\ $$$$\mathrm{case}\:\mathrm{1} \\ $$$${D}=\frac{{p}^{\mathrm{3}} }{\mathrm{27}}+\frac{{q}^{\mathrm{2}} }{\mathrm{4}}>\mathrm{0}\:\Rightarrow\:{x}^{\mathrm{3}} +{px}+{q}=\mathrm{0}\:\mathrm{has}\:\mathrm{got}\:\mathrm{1}\:\mathrm{real} \\ $$$$\mathrm{and}\:\mathrm{2}\:\mathrm{conjugated}\:\mathrm{complex}\:\mathrm{solutions}…
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Question Number 67762 by ugwu Kingsley last updated on 31/Aug/19 $${solve}\:{by}\:{the}\:{complex}\:{method} \\ $$$$ \\ $$$$ \\ $$$${y}^{{iv}} +\mathrm{3}{y}^{\mathrm{3}} +\mathrm{2}{y}^{\mathrm{2}} =−\mathrm{3}{sin}\mathrm{2}{x} \\ $$$$ \\ $$$$ \\…
Question Number 1895 by Yozzy last updated on 22/Oct/15 $${Let}\:{us}\:{generalise}\:{the}\:{result}\:{of}\:{taking}\:{the}\:{inverse}\:{tangent}\:{of}\:{a}\:{complex}\:{number} \\ $$$${to}\:{the}\:{form}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{tan}^{−\mathrm{1}} \left({c}+{id}\right)={a}+{ib} \\ $$$${where}\:{a},{b},{c},{d}\in\mathbb{R}\:{and}\:{i}=\sqrt{−\mathrm{1}}.\:{Determine}\:{a}\:{and}\:{b}\:{respectively}\:{in}\:{terms} \\ $$$${of}\:{c}\:{and}\:{d}.\: \\ $$ Commented by Rasheed Soomro…
Question Number 132937 by mohammad17 last updated on 17/Feb/21 $${solve}\:{the}\:{equation}\:{in}\:{complex}\:{number} \\ $$$${z}^{\mathrm{3}} ={z}_{{o}} \\ $$ Answered by mr W last updated on 17/Feb/21 $${z}_{\mathrm{0}} ={re}^{\theta{i}}…
Question Number 1559 by 123456 last updated on 19/Aug/15 $$\mathrm{find}\:\mathrm{complex}\:\mathrm{number}\:\alpha,\beta\:\mathrm{such}\:\mathrm{that} \\ $$$$\alpha^{{n}} =\beta^{{m}} \\ $$$$\beta^{{u}} =\alpha^{{v}} \\ $$$${n},{m},{u},{v}\in\mathbb{Z} \\ $$$$\boldsymbol{{Q}}\mathrm{1498} \\ $$ Answered by Rasheed…
Question Number 1540 by Rasheed Soomro last updated on 17/Aug/15 $$\mathrm{Determine}\:\mathrm{three}\:\mathrm{complex}\:\mathrm{numbers}\:\alpha\:,\:\beta\:,\gamma\:\:\mathrm{such}\:\mathrm{that} \\ $$$$\alpha=\beta^{\:\mathrm{2}} \:\:\:\:\:\:\:{but}\:\:\:\:\beta\:\neq\:\alpha^{\:\mathrm{2}} \\ $$$$\beta\:=\:\gamma^{\:\mathrm{2}} \:\:\:\:\:\:{but}\:\:\:\:\:\gamma\:\neq\:\beta^{\:\mathrm{2}} \\ $$$$\gamma\:=\:\alpha^{\:\mathrm{2}\:} \:\:\:\:\:{but}\:\:\:\:\:\alpha\:\neq\:\gamma^{\:\mathrm{2}} \\ $$ Answered by 123456…
Question Number 1498 by Rasheed Soomro last updated on 14/Aug/15 $$\mathrm{Find}\:\mathrm{complex}\:\mathrm{numbers}\:\alpha\:{and}\:\:\beta\:\:\mathrm{such}\:\mathrm{that} \\ $$$$\alpha^{\:{m}} =\beta^{\:\:{n}} \:\:\:\:{and}\:\:\:\beta^{\:\:{m}} =\alpha^{\mathrm{n}} \:\:,\:{m},{n}\:\in\:\mathbb{Z} \\ $$$$\boldsymbol{\mathrm{D}}\mathrm{etermine}\:\mathrm{formula}\:\mathrm{for}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{pairs}\:\left(\alpha,\beta\right)\:\mathrm{fulfilling}\:\mathrm{the} \\ $$$$\mathrm{above}\:\mathrm{conditions}.\:\:\left(\:\mathrm{You}\:\mathrm{may}\:\mathrm{ignore}\:\mathrm{this}\:\mathrm{part}\:\mathrm{in}\:\mathrm{your}\:\mathrm{answer}\right) \\ $$ Commented by…
Question Number 1497 by Rasheed Soomro last updated on 14/Aug/15 $$\mathrm{Find}\:\mathrm{complex}\:\mathrm{numbers}\:\alpha\:\mathrm{and}\:\beta\:\mathrm{such}\:\mathrm{that} \\ $$$$\alpha^{\mathrm{2}} =\beta^{\mathrm{3}} \:\:\mathrm{and}\:\:\beta^{\mathrm{2}} =\alpha^{\mathrm{3}} \\ $$ Commented by 123456 last updated on 14/Aug/15…
Question Number 1439 by Rasheed Ahmad last updated on 04/Aug/15 $${Are}\:{there}\:{two}\:{complex}\:{numbers} \\ $$$${which}\:{are}\:{cube}\:{of}\:{one}\:{another}? \\ $$$${If}\:{yes}\:{what}\:{are}\:{they}? \\ $$ Answered by 123456 last updated on 04/Aug/15 $$\begin{cases}{{x}={y}^{\mathrm{3}}…
Question Number 1426 by tabrez8590@gmail last updated on 31/Jul/15 $${why}\:{we}\:{can}\:{not}\:{compare}\:{any}\:{tow}\:{complex}\:{number}\:\: \\ $$$${how}\:{a}\:{cmplex}\:{number}\:{is}\:{uses}\:{a}\:{complex}\:{number}\:{practically}\:{please}\:{giveanexample} \\ $$ Commented by Rasheed Ahmad last updated on 03/Aug/15 $${Things}\:{with}\:{respect}\:{to}\:{single}\: \\ $$$${characteristic}\:{are}\:{easy}\:{to}\:{compare}.…