Solve the equation: sqrt((x-a)/x)+4sqrt(x/(x-a))=5?

I'd let t = sqrt( (x-a)/x ), then 1/t = sqrt(x/(x-a)).

You have t + 4/t = 5

Multiply by t, and you have t^2 + 4 = 5t, or t^2 -5t+4 = 0. Factor: (t-4)(t-1)=0.
One of the factors must be zero if the product is zero, so either t=4 or t=1.

When t=4, you have 4 = sqrt( (x-a)/x ), so 16x = x-a, and x = -a/15
When t=1, you have 1 = sqrt( (x-a)/x ), so x = x-a, and there is no solution.

Thus x = -a/15.

√((x - a)/x) + 4√(x/(x - a)) = 5
√((x - a)^2) + 4√(x^2) = 5√(x(x - a))
[(x - a) + 4x]^2 = 25x(x - a)
(5x - a)^2 = 25x(x - a)
25x^2 - 10ax + a^2 = 25x^2 - 25ax
a^2 = -15ax
a = -15x
x = -a/15

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