Solve for $x$: ​

I am going to try to simplify the denominators first.

The second denominator is  sqrt(6x) - 2·sqrt(3).

However, 2·sqrt(3) can be written as sqrt(4)·sqrt(3) = sqrt(12)

     and sqrt(12) can be written as sqrt(6)·sqrt(2),

so sqrt(6x) - 2·sqrt(3) becomes sqrt(6)·sqrt(x) - sqrt(6)·sqrt(2),

and, factoring, becomes:  sqrt(6)[sqrt(x) - sqrt(2)].

This is almost the same as the first denominator, which is simply [sqrt(x) - sqrt(2)].

This means that if x = 2, these denominators are zero, so if we get an answer of 2, we'll have to throw it out.

Now, let's multiply both sides by sqrt(6)[sqrt(x) - sqrt(2)].

This cancels the denominator on the left side of the equation, but puts a factor of sqrt(6) on the left side.

This completely cancels the denominator on the right side of the equation.

We now have:  sqrt(6)·[sqrt(3x) - 4sqrt(3)]  =  2sqrt(2x) + sqrt(2)

Let's square both sides:

Left side:     [ sqrt(6)·[sqrt(3x) - 4sqrt(3)] ]²  = 6·[ sqrt(3x) - 4sqrt(3) ]²  =  6[ 3x - 2·4·sqrt(3x)·sqrt(3) + 16·3 ]

     =  6[3x - 8·3·sqrt(x) + 48]  =  18x - 144sqrt(x) + 288

Right side:     [ 2sqrt(2x) + sqrt(2) ]²  =  4·2x + 2·2·2·sqrt(x) + 2  

     =  8x + 8sqrt(x) + 2

Setting these two equal:     18x - 144sqrt(x) + 288  =  8x + 8sqrt(x) + 2

Simplifying:     10x - 152sqrt(x) + 286 = 0

Dividing by 2:     5x - 76sqrt(x) + 143  =  0

Using the quadratic formula to find sqrt(x);     sqrt(x)  =  [ 76 +/1 sqrt(76² - 4·5·143) ] / 10

----->     sqrt(x)  =  13 or 2.2

----->     x = 169  or  x = 4.84

Checking both possible answers, 169 works but 4.84 is an extraneous root introduced by the squaring process.

[ √[3x] - 4√3 ]   =   2√[2x] + √2

__________       __________

√x  - √2                 √[6x] - 2√3

[ √3 (√x - 4) ]  =  [ √2 ( 2√x + 1) ]

__________     _____________ 

√x  - √2               √[6x] -  √12 

[ √3 (√x - 4) ]  =  [ √2 ( 2√x + 1) ]

__________     _____________

√x  - √2               √6 [ √x - √2 ]

[ √3 (√x - 4) ]  =  [ √2 ( 2√x + 1) ]

                           _____________

                               √6

[ √3 (√x - 4) ]  =  [ ( 2√x + 1) ] / √3

3[√x -  4]    =     2√x + 1

3√x - 12   =   2√x + 1

√x  =  13

x  = 169

Solve for x:
(sqrt(3) sqrt(x) - 4 sqrt(3))/(sqrt(x) - sqrt(2)) = (2 sqrt(2) sqrt(x) + sqrt(2))/(sqrt(6) sqrt(x) - 2 sqrt(3))

Cross multiply:
(sqrt(3) sqrt(x) - 4 sqrt(3)) (sqrt(6) sqrt(x) - 2 sqrt(3)) = (sqrt(x) - sqrt(2)) (2 sqrt(2) sqrt(x) + sqrt(2))

Subtract (sqrt(x) - sqrt(2)) (2 sqrt(2) sqrt(x) + sqrt(2)) from both sides:
(sqrt(3) sqrt(x) - 4 sqrt(3)) (sqrt(6) sqrt(x) - 2 sqrt(3)) - (sqrt(x) - sqrt(2)) (sqrt(2) + 2 sqrt(2) sqrt(x)) = 0

(sqrt(3) sqrt(x) - 4 sqrt(3)) (sqrt(6) sqrt(x) - 2 sqrt(3)) - (sqrt(x) - sqrt(2)) (sqrt(2) + 2 sqrt(2) sqrt(x)) = 26 + (-13 sqrt(2) - 2) sqrt(x) + sqrt(2) x:
26 + (-13 sqrt(2) - 2) sqrt(x) + sqrt(2) x = 0

Simplify and substitute y = sqrt(x).
26 + (-13 sqrt(2) - 2) sqrt(x) + sqrt(2) x = 26 + (-13 sqrt(2) - 2) sqrt(x) + sqrt(2) (sqrt(x))^2
 = sqrt(2) y^2 + (-2 - 13 sqrt(2)) y + 26:
sqrt(2) y^2 + (-2 - 13 sqrt(2)) y + 26 = 0

The left hand side factors into a product with two terms:
(y - 13) (sqrt(2) y - 2) = 0

Split into two equations:
y - 13 = 0 or sqrt(2) y - 2 = 0

Add 13 to both sides:
y = 13 or sqrt(2) y - 2 = 0

Substitute back for y = sqrt(x):
sqrt(x) = 13 or sqrt(2) y - 2 = 0

Raise both sides to the power of two:
x = 169 or sqrt(2) y - 2 = 0

Add 2 to both sides:
x = 169 or sqrt(2) y = 2

Divide both sides by sqrt(2):
x = 169 or y = sqrt(2)

Substitute back for y = sqrt(x):
x = 169 or sqrt(x) = sqrt(2)

Raise both sides to the power of two:
x = 169 or x = 2

(sqrt(3) sqrt(x) - 4 sqrt(3))/(sqrt(x) - sqrt(2)) ⇒ (sqrt(3) sqrt(2) - 4 sqrt(3))/(sqrt(2) - sqrt(2)) = ∞^~
(2 sqrt(2) sqrt(x) + sqrt(2))/(sqrt(6) sqrt(x) - 2 sqrt(3)) ⇒ (sqrt(2) + 2 sqrt(2) sqrt(2))/(sqrt(6) sqrt(2) - 2 sqrt(3)) = ∞^~:
So this solution is incorrect

(sqrt(3) sqrt(x) - 4 sqrt(3))/(sqrt(x) - sqrt(2)) ⇒ (sqrt(3) sqrt(169) - 4 sqrt(3))/(sqrt(169) - sqrt(2)) = -(9 sqrt(3))/(sqrt(2) - 13) ≈ 1.34548
(2 sqrt(2) sqrt(x) + sqrt(2))/(sqrt(6) sqrt(x) - 2 sqrt(3)) ⇒ (sqrt(2) + 2 sqrt(2) sqrt(169))/(sqrt(6) sqrt(169) - 2 sqrt(3)) = (9 sqrt(6))/(13 sqrt(2) - 2) ≈ 1.34548:
So this solution is correct

x = 169 - Courtesy of Mathematica 11 - !!!!