12x^2+24x+8=0

Solution for 12x^2+24x+8=0 equation:

Simplifying
12x2 + 24x + 8 = 0

Reorder the terms:
8 + 24x + 12x2 = 0

Solving
8 + 24x + 12x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '4'.
4(2 + 6x + 3x2) = 0

Ignore the factor 4.
Subproblem 1
Set the factor '(2 + 6x + 3x2)' equal to zero and attempt to solve:

Simplifying
2 + 6x + 3x2 = 0

Solving
2 + 6x + 3x2 = 0

Begin completing the square.  Divide all terms by
3 the coefficient of the squared term: 

Divide each side by '3'.
0.6666666667 + 2x + x2 = 0

Move the constant term to the right:

Add '-0.6666666667' to each side of the equation.
0.6666666667 + 2x + -0.6666666667 + x2 = 0 + -0.6666666667

Reorder the terms:
0.6666666667 + -0.6666666667 + 2x + x2 = 0 + -0.6666666667

Combine like terms: 0.6666666667 + -0.6666666667 = 0.0000000000
0.0000000000 + 2x + x2 = 0 + -0.6666666667
2x + x2 = 0 + -0.6666666667

Combine like terms: 0 + -0.6666666667 = -0.6666666667
2x + x2 = -0.6666666667

The x term is 2x.  Take half its coefficient (1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
2x + 1 + x2 = -0.6666666667 + 1

Reorder the terms:
1 + 2x + x2 = -0.6666666667 + 1

Combine like terms: -0.6666666667 + 1 = 0.3333333333
1 + 2x + x2 = 0.3333333333

Factor a perfect square on the left side:
(x + 1)(x + 1) = 0.3333333333

Calculate the square root of the right side: 0.577350269

Break this problem into two subproblems by setting 
(x + 1) equal to 0.577350269 and -0.577350269.

Subproblem 1
x + 1 = 0.577350269

Simplifying
x + 1 = 0.577350269

Reorder the terms:
1 + x = 0.577350269

Solving
1 + x = 0.577350269

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-1' to each side of the equation.
1 + -1 + x = 0.577350269 + -1

Combine like terms: 1 + -1 = 0
0 + x = 0.577350269 + -1
x = 0.577350269 + -1

Combine like terms: 0.577350269 + -1 = -0.422649731
x = -0.422649731

Simplifying
x = -0.422649731

Subproblem 2
x + 1 = -0.577350269

Simplifying
x + 1 = -0.577350269

Reorder the terms:
1 + x = -0.577350269

Solving
1 + x = -0.577350269

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-1' to each side of the equation.
1 + -1 + x = -0.577350269 + -1

Combine like terms: 1 + -1 = 0
0 + x = -0.577350269 + -1
x = -0.577350269 + -1

Combine like terms: -0.577350269 + -1 = -1.577350269
x = -1.577350269

Simplifying
x = -1.577350269

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.422649731, -1.577350269}
Solution
x = {-0.422649731, -1.577350269}