The domain of this equation is the set { x | x >= 3/4 }.

It is the set of x, where the expression under the square root is non-negative: 4x - 3 >= 0.


Next, introduce new variable  u = .
Notice that u always is non-negative.


Now, the original equation takes the form

    u + 40/u = 12x + 14,

or

    u + 40/u = 3*(4x-3) + 23

    u + 40/u = 3u^2 + 23.


Multiply both sides by u

    u^2 + 40 = 3u^3 + 23u

    3u^3 - u^2 + 23u - 40 = 0.


This equation has NO rational roots that can be found using Rational Root test.


There is a unique real root u = 1.43989 (rounded).


So,   = 1.43989,  4x-3 = 1.43989^2 = 2.073283212,  x =  = 1.268320803.


This value is in the domain, so the unique real solution to the given equation is 1.268320803 (approximately).