Question 1209597
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Find all values of x satisfying

sqrt(4x - 3) + 40/sqrt(4x - 3) = 12x + 14.
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<pre>
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 = {{{sqrt(4x-3)}}}.
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,  {{{sqrt(4x-3)}}} = 1.43989,  4x-3 = 1.43989^2 = 2.073283212,  x = {{{(2.073283212+3)/4}}} = 1.268320803.


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

Solved.


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Another method is to solve it using graphing calculators or numerical methods.


For the graphic solution, see the plot under this link


<A HREF=https://www.desmos.com/calculator/wmrhakfyrj>https://www.desmos.com/calculator/wmrhakfyrj</A>


https://www.desmos.com/calculator/wmrhakfyrj


Click on the intersection point to see the coordinates of the intersection point.