Find all values of satisfying ​

Solve     sqrt(4x - 3) + 10 / sqrt(4x - 3)  =  7

First note that the expression 4x - 3 is under the square root sign.

Since this value must be positive:      4x - 3  >=  0     --->     4x  >= 3     --->     x >= 4/3

To remove the denominator from the problem multiply each side by  sqrt(4x - 3):

     sqrt(4x - 3) [ sqrt(4x - 3) + 10 / sqrt(4x - 3) ]    =    sqrt(4x - 3) [ 7 ]

--->     sqrt(4x - 3) · sqrt(4x - 3)  +  sqrt(4x - 3) · 10  sqrt(4x - 3)    =    7· sqrt(4x - 3)

--->                (4x - 3)                  +                  10                            =    7 · sqrt(4x - 3)

--->                                     4x + 7   =   7·sqrt(4x - 3)

Square both sides:             [ 4x + 7 ]²   =   [ 7·sqrt(4x - 3) ]²  

                                   4x² + 56x + 49   =   49(4x - 3)

--->                             4x² + 56x + 49   =   196x - 147

Subtract 196x from both sides, add 147 to both sides:

--->                          4x² - 140x + 196   =   0

Divide both sides by 4:

--->                            x² - 35x + 49   =   0

Factor:                      (4x - 7)(x - 7)  =  0

So, either 4x - 7  =  0   --->    x = 7/4     or     x - 7  =  0     --->   x = 7

Since both answers are > 4/3, these are possible answer.

I'll leave the check for you ...

Let  √[4x - 3]  =  q^2

So we have

q^2  +  10/q^2   = 7      multiply both sides by q^2

q^4 + 10  = 7q^2

q^4 - 7q^2 + 10   = 0    factor

(q^2 - 5) ( q^2 - 2)  = 0

So either

q^2  = 5     or    q^2   =  2    .....so....

√[4x - 3]  = 5        square both sides

4x - 3   = 25     add 3 to both sides

4x = 28

x = 7

Or

√[4x - 3]  = 2     square both sides

4x - 3  = 4      add 3 to both sides

4x = 7

x = 7/4

Check  x = 7

√[4x - 3] +  10 / √[4x - 3] = 7  ??

√[4*7 - 3] + 10 / √[4*7 - 3]  = 7  ??

5 + 10/5  = 7      is true

Check  x  = 7/4

√[4(7/4) - 3] +  10 / √[4(7/4) - 3] = 7   ??

√[7-3] +  10 / √[7 - 3]] = 7  ??

2 + 10/2   = 7   is also true

So  x   =  7    or   x  = 7/4