Problem with solving a complex equation












1












$begingroup$


I'm having trouble solving this equation
$$Z^3=-4ibar Z$$
I need to find Z, I've tried multiplying the equation by Z but still couldnt solve it.



I'll be glad for help.






algebra-precalculus complex-numbers







share|cite|improve this question











$endgroup$

















    1












    $begingroup$


    I'm having trouble solving this equation
    $$Z^3=-4ibar Z$$
    I need to find Z, I've tried multiplying the equation by Z but still couldnt solve it.



    I'll be glad for help.






    algebra-precalculus complex-numbers







    share|cite|improve this question











    $endgroup$















      1












      1








      1





      $begingroup$


      I'm having trouble solving this equation
      $$Z^3=-4ibar Z$$
      I need to find Z, I've tried multiplying the equation by Z but still couldnt solve it.



      I'll be glad for help.






      algebra-precalculus complex-numbers







      share|cite|improve this question











      $endgroup$




      I'm having trouble solving this equation
      $$Z^3=-4ibar Z$$
      I need to find Z, I've tried multiplying the equation by Z but still couldnt solve it.



      I'll be glad for help.






      algebra-precalculus complex-numbers




      algebra-precalculus complex-numbers






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Dec 28 '18 at 5:32









      Eric Wofsey

      187k14215344




      187k14215344










      asked Feb 26 '18 at 16:40









      user534957user534957

      275




      275






















          3 Answers
          3






          active

          oldest

          votes


















          1












          $begingroup$

          Note that



          $$Z^3=-4ibar Zimplies |Z^3|=4|Z|implies |Z|=0 quad lor quad |Z|=2$$



          and for $|Z|=2$



          $$Z^3=-4ibar Ziff Z^4=-4iZbar Z=-16i implies Z=2(-i)^frac14$$






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Thank you for your comment!
            $endgroup$
            – user534957
            Feb 26 '18 at 16:52










          • $begingroup$
            Can i ask how did you calculate |Z^3|=4|Z|?
            $endgroup$
            – user534957
            Feb 26 '18 at 16:53










          • $begingroup$
            I get it. Very nice thank you so much!!
            $endgroup$
            – user534957
            Feb 26 '18 at 16:55










          • $begingroup$
            @user534957 I've taken the modulus both sides.
            $endgroup$
            – gimusi
            Feb 26 '18 at 16:55



















          2












          $begingroup$

          I'd try taking the modulus of both sides: $|Z|^3=4|Z|$, from which $|Z| = 0$ or $2$.



          For the case $|Z|=2$, we have $Z=2e^{itheta}$.



          Can you progress from there?






          share|cite|improve this answer









          $endgroup$





















            0












            $begingroup$

            HINT.



            $$Z = X + iY$$



            $$bar Z = X - iY$$



            Assuming the "bar" symbol denotes the complex conjugate.



            If you multiply by $Z$ you get



            $$(X + iY)^4 = 4i(X^2 + Y^2)$$



            $$(X + iY)^4 = 4i (X + iY)(X - iY)$$



            $$(X + iY)^3 = 4i(X - iY)$$






            share|cite|improve this answer









            $endgroup$













            • $begingroup$
              How do I find z with this?
              $endgroup$
              – user534957
              Feb 26 '18 at 16:46










            • $begingroup$
              And thank you for your cimment i really appreciate it
              $endgroup$
              – user534957
              Feb 26 '18 at 16:47











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            3 Answers
            3






            active

            oldest

            votes








            3 Answers
            3






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes









            1












            $begingroup$

            Note that



            $$Z^3=-4ibar Zimplies |Z^3|=4|Z|implies |Z|=0 quad lor quad |Z|=2$$



            and for $|Z|=2$



            $$Z^3=-4ibar Ziff Z^4=-4iZbar Z=-16i implies Z=2(-i)^frac14$$






            share|cite|improve this answer









            $endgroup$













            • $begingroup$
              Thank you for your comment!
              $endgroup$
              – user534957
              Feb 26 '18 at 16:52










            • $begingroup$
              Can i ask how did you calculate |Z^3|=4|Z|?
              $endgroup$
              – user534957
              Feb 26 '18 at 16:53










            • $begingroup$
              I get it. Very nice thank you so much!!
              $endgroup$
              – user534957
              Feb 26 '18 at 16:55










            • $begingroup$
              @user534957 I've taken the modulus both sides.
              $endgroup$
              – gimusi
              Feb 26 '18 at 16:55
















            1












            $begingroup$

            Note that



            $$Z^3=-4ibar Zimplies |Z^3|=4|Z|implies |Z|=0 quad lor quad |Z|=2$$



            and for $|Z|=2$



            $$Z^3=-4ibar Ziff Z^4=-4iZbar Z=-16i implies Z=2(-i)^frac14$$






            share|cite|improve this answer









            $endgroup$













            • $begingroup$
              Thank you for your comment!
              $endgroup$
              – user534957
              Feb 26 '18 at 16:52










            • $begingroup$
              Can i ask how did you calculate |Z^3|=4|Z|?
              $endgroup$
              – user534957
              Feb 26 '18 at 16:53










            • $begingroup$
              I get it. Very nice thank you so much!!
              $endgroup$
              – user534957
              Feb 26 '18 at 16:55










            • $begingroup$
              @user534957 I've taken the modulus both sides.
              $endgroup$
              – gimusi
              Feb 26 '18 at 16:55














            1












            1








            1





            $begingroup$

            Note that



            $$Z^3=-4ibar Zimplies |Z^3|=4|Z|implies |Z|=0 quad lor quad |Z|=2$$



            and for $|Z|=2$



            $$Z^3=-4ibar Ziff Z^4=-4iZbar Z=-16i implies Z=2(-i)^frac14$$






            share|cite|improve this answer









            $endgroup$



            Note that



            $$Z^3=-4ibar Zimplies |Z^3|=4|Z|implies |Z|=0 quad lor quad |Z|=2$$



            and for $|Z|=2$



            $$Z^3=-4ibar Ziff Z^4=-4iZbar Z=-16i implies Z=2(-i)^frac14$$







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered Feb 26 '18 at 16:47









            gimusigimusi

            92.8k84494




            92.8k84494












            • $begingroup$
              Thank you for your comment!
              $endgroup$
              – user534957
              Feb 26 '18 at 16:52










            • $begingroup$
              Can i ask how did you calculate |Z^3|=4|Z|?
              $endgroup$
              – user534957
              Feb 26 '18 at 16:53










            • $begingroup$
              I get it. Very nice thank you so much!!
              $endgroup$
              – user534957
              Feb 26 '18 at 16:55










            • $begingroup$
              @user534957 I've taken the modulus both sides.
              $endgroup$
              – gimusi
              Feb 26 '18 at 16:55


















            • $begingroup$
              Thank you for your comment!
              $endgroup$
              – user534957
              Feb 26 '18 at 16:52










            • $begingroup$
              Can i ask how did you calculate |Z^3|=4|Z|?
              $endgroup$
              – user534957
              Feb 26 '18 at 16:53










            • $begingroup$
              I get it. Very nice thank you so much!!
              $endgroup$
              – user534957
              Feb 26 '18 at 16:55










            • $begingroup$
              @user534957 I've taken the modulus both sides.
              $endgroup$
              – gimusi
              Feb 26 '18 at 16:55
















            $begingroup$
            Thank you for your comment!
            $endgroup$
            – user534957
            Feb 26 '18 at 16:52




            $begingroup$
            Thank you for your comment!
            $endgroup$
            – user534957
            Feb 26 '18 at 16:52












            $begingroup$
            Can i ask how did you calculate |Z^3|=4|Z|?
            $endgroup$
            – user534957
            Feb 26 '18 at 16:53




            $begingroup$
            Can i ask how did you calculate |Z^3|=4|Z|?
            $endgroup$
            – user534957
            Feb 26 '18 at 16:53












            $begingroup$
            I get it. Very nice thank you so much!!
            $endgroup$
            – user534957
            Feb 26 '18 at 16:55




            $begingroup$
            I get it. Very nice thank you so much!!
            $endgroup$
            – user534957
            Feb 26 '18 at 16:55












            $begingroup$
            @user534957 I've taken the modulus both sides.
            $endgroup$
            – gimusi
            Feb 26 '18 at 16:55




            $begingroup$
            @user534957 I've taken the modulus both sides.
            $endgroup$
            – gimusi
            Feb 26 '18 at 16:55











            2












            $begingroup$

            I'd try taking the modulus of both sides: $|Z|^3=4|Z|$, from which $|Z| = 0$ or $2$.



            For the case $|Z|=2$, we have $Z=2e^{itheta}$.



            Can you progress from there?






            share|cite|improve this answer









            $endgroup$


















              2












              $begingroup$

              I'd try taking the modulus of both sides: $|Z|^3=4|Z|$, from which $|Z| = 0$ or $2$.



              For the case $|Z|=2$, we have $Z=2e^{itheta}$.



              Can you progress from there?






              share|cite|improve this answer









              $endgroup$
















                2












                2








                2





                $begingroup$

                I'd try taking the modulus of both sides: $|Z|^3=4|Z|$, from which $|Z| = 0$ or $2$.



                For the case $|Z|=2$, we have $Z=2e^{itheta}$.



                Can you progress from there?






                share|cite|improve this answer









                $endgroup$



                I'd try taking the modulus of both sides: $|Z|^3=4|Z|$, from which $|Z| = 0$ or $2$.



                For the case $|Z|=2$, we have $Z=2e^{itheta}$.



                Can you progress from there?







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Feb 26 '18 at 16:45









                paw88789paw88789

                29.2k12349




                29.2k12349























                    0












                    $begingroup$

                    HINT.



                    $$Z = X + iY$$



                    $$bar Z = X - iY$$



                    Assuming the "bar" symbol denotes the complex conjugate.



                    If you multiply by $Z$ you get



                    $$(X + iY)^4 = 4i(X^2 + Y^2)$$



                    $$(X + iY)^4 = 4i (X + iY)(X - iY)$$



                    $$(X + iY)^3 = 4i(X - iY)$$






                    share|cite|improve this answer









                    $endgroup$













                    • $begingroup$
                      How do I find z with this?
                      $endgroup$
                      – user534957
                      Feb 26 '18 at 16:46










                    • $begingroup$
                      And thank you for your cimment i really appreciate it
                      $endgroup$
                      – user534957
                      Feb 26 '18 at 16:47
















                    0












                    $begingroup$

                    HINT.



                    $$Z = X + iY$$



                    $$bar Z = X - iY$$



                    Assuming the "bar" symbol denotes the complex conjugate.



                    If you multiply by $Z$ you get



                    $$(X + iY)^4 = 4i(X^2 + Y^2)$$



                    $$(X + iY)^4 = 4i (X + iY)(X - iY)$$



                    $$(X + iY)^3 = 4i(X - iY)$$






                    share|cite|improve this answer









                    $endgroup$













                    • $begingroup$
                      How do I find z with this?
                      $endgroup$
                      – user534957
                      Feb 26 '18 at 16:46










                    • $begingroup$
                      And thank you for your cimment i really appreciate it
                      $endgroup$
                      – user534957
                      Feb 26 '18 at 16:47














                    0












                    0








                    0





                    $begingroup$

                    HINT.



                    $$Z = X + iY$$



                    $$bar Z = X - iY$$



                    Assuming the "bar" symbol denotes the complex conjugate.



                    If you multiply by $Z$ you get



                    $$(X + iY)^4 = 4i(X^2 + Y^2)$$



                    $$(X + iY)^4 = 4i (X + iY)(X - iY)$$



                    $$(X + iY)^3 = 4i(X - iY)$$






                    share|cite|improve this answer









                    $endgroup$



                    HINT.



                    $$Z = X + iY$$



                    $$bar Z = X - iY$$



                    Assuming the "bar" symbol denotes the complex conjugate.



                    If you multiply by $Z$ you get



                    $$(X + iY)^4 = 4i(X^2 + Y^2)$$



                    $$(X + iY)^4 = 4i (X + iY)(X - iY)$$



                    $$(X + iY)^3 = 4i(X - iY)$$







                    share|cite|improve this answer












                    share|cite|improve this answer



                    share|cite|improve this answer










                    answered Feb 26 '18 at 16:43









                    Von NeumannVon Neumann

                    16.4k72545




                    16.4k72545












                    • $begingroup$
                      How do I find z with this?
                      $endgroup$
                      – user534957
                      Feb 26 '18 at 16:46










                    • $begingroup$
                      And thank you for your cimment i really appreciate it
                      $endgroup$
                      – user534957
                      Feb 26 '18 at 16:47


















                    • $begingroup$
                      How do I find z with this?
                      $endgroup$
                      – user534957
                      Feb 26 '18 at 16:46










                    • $begingroup$
                      And thank you for your cimment i really appreciate it
                      $endgroup$
                      – user534957
                      Feb 26 '18 at 16:47
















                    $begingroup$
                    How do I find z with this?
                    $endgroup$
                    – user534957
                    Feb 26 '18 at 16:46




                    $begingroup$
                    How do I find z with this?
                    $endgroup$
                    – user534957
                    Feb 26 '18 at 16:46












                    $begingroup$
                    And thank you for your cimment i really appreciate it
                    $endgroup$
                    – user534957
                    Feb 26 '18 at 16:47




                    $begingroup$
                    And thank you for your cimment i really appreciate it
                    $endgroup$
                    – user534957
                    Feb 26 '18 at 16:47


















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