x+6 = 6*sqrt(x-2)

First let us square both sides:

==? (x+6)^2 = [6*sqrt(x-2)]^2

==> x^2 + 12x + 36 = 36*(x-2)

==> x^2 + 12x + 36 = 36x - 72

Group similar:

==> x^2 - 24x - 108 = 0

Now let us factor:

==> (x-18)(x+6) = 0

==> x1= 18

==> x2= -6 (impossible solution because sqrt(x-2) is not defined.

The the answer is x = 18

For the beginning, we'll impose the constraint of existence of the square root:

x-2>0

x>2

The admissible solutions of the equation have to belong to the interval (2,+inf.).

6*sqrt(x-2)=x+6

We'll square raise the expression above:

36(x-2)=x^2+12x+36

36x-72=x^2+12x+36

x^2-24x+108=0

We'll apply the quadratic formula:

x1=[24+ sqrt (24^2-4*108)]/2=18

x2=[24- sqrt (24^2-4*108)]/2=6

x1= 18>2

x2 = 6>2

Since both solutions belong to the interval (2,+inf.), they are admissible.

x+6 = 6*sqrt(x-2).

To solve for x:

We square both sides of the given equation to get rid of square root sign.

(x+6)^2 = 6^2 (x-2)

x^2+12x+36 = 36(x-2)

x^2+12x+36 -36(x-2) = 0

x^2+(12-36)x+36 +72 = 0

x^2 -24x +108 = 0

x^2-18x -6x + 108 = 0

x(x-18) +6(x-18) = 0

(x-18)(x+6) = 0

x-18 = 0 or x+6 = 0

x=-18 or x = -6

Given:

x + 6 = 6*sqrt(x - 2)

Taking square of both the sides:

(x + 6)^2 = (6^2)(x - 2)

--> x^2 + 12x + 36 = 36x - 72

--> x^2 - 24x + 108 = 0

--> x^2 - 6x - 18x + 108 = 0

--> x(x - 6) - 18(x - 6) = 0

--> (x - 6)(x -18) = 0

Therefore x has two possible values:

x = 6 and 8