Question 1130004: Solve the equation (y+5/y)^2 + 3(y+5/y)=4 ,using substitution u=y+5/y
Answer by ikleyn(52775) (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 = , 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.
Solved and explained.
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