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

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Question 1130004: Solve the equation (y+5/y)^2 + 3(y+5/y)=4 ,using substitution u=y+5/y
Answer by ikleyn(52775) About Me  (Show Source):
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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+%2B+5%2Fy, 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+%2B+5%2Fy = -4  and  y+%2B+5%2Fy = 1 to find possible solutions for "y".


1)  y+%2B+5%2Fy = -4  <====> is equivalent to

    y%5E2+%2B+4y+%2B+5 = 0  <=====>  is equivalent to

    %28y+%2B+2%29%5E2+%2B+1 = 0   or   %28y%2B2%29%5E2 = -1

                              which has no solutions in real numbers,

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



2)  y+%2B+5%2Fy = 1  <====> is equivalent to

    y%5E2+-+y+%2B+5 = 0  <=====>  is equivalent to

    %28y+-+0.5%29%5E2+%2B+4.75 = 0   or   %28y-0.5%29%5E2 = -4.75,

                              which has no solutions in real numbers,

    but has two solutions in complex numbers  x= 0.5+%2B+i%2Asqrt%284.75%29  and  x= 0.5+-i%2Asqrt%284.75%29.


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

Solved and explained.