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Solve `Sqrt(X/(X- 3)) + Sqrt((X - 3)/X) = 5/2` - Mathematics

Solve `sqrt(x/(x- 3)) + sqrt((x - 3)/x) = 5/2`

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Solution

`sqrt(x/(x- 3)) + sqrt((x - 3)/x) = 5/2`

Let  `sqrt(x/(x-3)) = y`

Then `y + 1/y = 5/2`

`=> (y^2 + 1)/y = 5/2`

`=> 2y^2 + 2 = 5y`

`=> 2y^2 - 5y + 2 = 0`

`=> 2y^2 - 4y - y + 2 = 0`

=> (y - 2)(2y - 1) = 0

if y - 2 = 0  or 2y - 1 = 0

then y = 2  or y = 1/2

`=> sqrt(x/(x - 3)) = 2` or `sqrt(x/(x - 3)) = 1/2`

`=> x/(x - 3) = 4 or x/(x - 3) = 1/4`

=> x = 4 or x = -1

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