The equation to be solved using completion of squares is 6x^2 + 36x + 18 = 0.

6x^2 + 36x + 18 = 0

=> x^2 + 6x + 3 = 0

=> x^2 + 6x + 9 = 6

=> (x + 3)^2 = 6

We now have x + 3 = sqrt 6 and x + 3 = -sqrt 6

=> x = -3 + sqrt 6 and x = -3 - sqrt 6

**The roots of the given equation are x = -3 + sqrt 6 and x = -3 - sqrt 6**

The equation 6x^2 + 36x + 18 = 0 has to be solved.

6x^2 + 36x + 18 = 0

Divide by 6

x^2 + 6x + 3 = 0

x^2 + 6x + 9 + 3 = 9

(x + 3)^2 + 3 = 9

(x + 3)^2 = 9 - 3

(x + 3)^2 = 6

x + 3 = `+- sqrt 6`

x = `-3 +- sqrt 6`

The solution of the equation is x = `-3 +- sqrt 6`

We'll divide by 6 entire equation:

x^2 + 6x + 3 = 0

Now, we'll shift 3 to the right to become much more easy to see what has to be added to complete the square:

x^2 + 6x = -3

We'll consider the formula:

(a+b)^2 = a^2 + 2ab + b^2

We'll identify the first 2 terms:

a^2 = x^2 and 2*a*b = 2*x*3

The missing term is b and we'll identify it as being 3.

We'll raise to square 3 and we'll add it both sides:

x^2 + 6x + 9 = -3 + 9

(x+3)^2 = 6

We'll take the square root both sides:

x + 3 = sqrt 6

x = -3 + sqrt 6

x + 3 = - sqrt 6

x = -3 - sqrt 6

**The roots of quadratic have been determined and they are:{-3 - sqrt 6 ; -3 + sqrt 6}.**