Question 1130004
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Solve the equation (y+5/y)^2 + 3(y+5/y)=4 ,using substitution u=y+5/y
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When you introduce new variable  u = {{{y + 5/y}}}, 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  {{{y + 5/y}}} = -4  and  {{{y + 5/y}}} = 1 to find possible solutions for "y".


1)  {{{y + 5/y}}} = -4  <====> is equivalent to

    {{{y^2 + 4y + 5}}} = 0  <=====>  is equivalent to

    {{{(y + 2)^2 + 1}}} = 0   or   {{{(y+2)^2}}} = -1

                              which has no solutions in real numbers,

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



2)  {{{y + 5/y}}} = 1  <====> is equivalent to

    {{{y^2 - y + 5}}} = 0  <=====>  is equivalent to

    {{{(y - 0.5)^2 + 4.75}}} = 0   or   {{{(y-0.5)^2}}} = -4.75,

                              which has no solutions in real numbers,

    but has two solutions in complex numbers  x= {{{0.5 + i*sqrt(4.75)}}}  and  x= {{{0.5 -i*sqrt(4.75)}}}.


<U>Answer</U>.  Your original equation has no solution/solutions in real numbers, but has 4 (four) solutions in complex numbers, listed above.
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Solved and explained.