Question

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Solve:
$\sf \sqrt{ \dfrac{x}{x - 3} } + \sqrt{ \dfrac{x - 3}{x} } = \dfrac{5}{3}$
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Answer 1

Answer:

The solution is given in the above image

Note Your question is wrong in the denominator I have replaced 3 by 2

Step-by-step explanation:

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Answer 2

Let √(x)/(x - 3) = t , so, √(x - 3)/x = 1/t

Therefore,

=> t + 1/t = 5/3

=> (t² + 1)/t = 5/3

=> 3(t² + 1) = 5t

=> 3t² - 5t + 3 = 0

Notice that the discriminant of this equation is (-5)^2 - 4(3)(3) = - 11 < 0 , it means it has no real solution and no real value of t satisfies this.

Moreover, in complex numbers:

t = (-(-5) ± √-11)/3(2) = (5 ± √11i)/6

Hence,

x/(x - 3) = t² = ((5 ± √11i)/6)²

x/(x - 3) = (7 ± 5√11i)/18

18x = (x - 3)(7 ± 5√11i)

18x = 7x ± 5xi√11 - 21 ± 15i√11

11x ± 5xi√11 = ± 15i√11 - 21

x(11 ± 5√11i) = (±15i√11 - 21)

x = (±15i√11 - 21)/(11 ± 5√11i)

You can solve this further, rationalize the denominator. On solving,

x = (3/2) ± (15√11i)/22