SOLUTION: Solve the equation (y+5/y)^2 + 3(y+5/y)=4 ,using substitution u=y+5/y

When you introduce new variable  u = , the original equation takes the form


u^2 + 3u - 4 = 0.


Factoring, you get


(u+4)*(u-1) = 0,


which gives you two roots  u= -4  and  u= 1.


Thus, now you need to solve two equations   = -4  and   = 1 to find possible solutions for "y".


1)   = -4  <====> is equivalent to

     = 0  <=====>  is equivalent to

     = 0   or    = -1

                              which has no solutions in real numbers,

    but has two solutions in complex numbers  x= -2 + i  and  x= -2 -i.  



2)   = 1  <====> is equivalent to

     = 0  <=====>  is equivalent to

     = 0   or    = -4.75,

                              which has no solutions in real numbers,

    but has two solutions in complex numbers  x=   and  x= .


Answer.  Your original equation has no solution/solutions in real numbers, but has 4 (four) solutions in complex numbers, listed above.