12x^2-24x-6=0

Solution for 12x^2-24x-6=0 equation:

Simplifying
12x2 + -24x + -6 = 0

Reorder the terms:
-6 + -24x + 12x2 = 0

Solving
-6 + -24x + 12x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '6'.
6(-1 + -4x + 2x2) = 0

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

Simplifying
-1 + -4x + 2x2 = 0

Solving
-1 + -4x + 2x2 = 0

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

Divide each side by '2'.
-0.5 + -2x + x2 = 0

Move the constant term to the right:

Add '0.5' to each side of the equation.
-0.5 + -2x + 0.5 + x2 = 0 + 0.5

Reorder the terms:
-0.5 + 0.5 + -2x + x2 = 0 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + -2x + x2 = 0 + 0.5
-2x + x2 = 0 + 0.5

Combine like terms: 0 + 0.5 = 0.5
-2x + x2 = 0.5

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 = 0.5 + 1

Reorder the terms:
1 + -2x + x2 = 0.5 + 1

Combine like terms: 0.5 + 1 = 1.5
1 + -2x + x2 = 1.5

Factor a perfect square on the left side:
(x + -1)(x + -1) = 1.5

Calculate the square root of the right side: 1.224744871

Break this problem into two subproblems by setting 
(x + -1) equal to 1.224744871 and -1.224744871.

Subproblem 1
x + -1 = 1.224744871

Simplifying
x + -1 = 1.224744871

Reorder the terms:
-1 + x = 1.224744871

Solving
-1 + x = 1.224744871

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 = 1.224744871 + 1

Combine like terms: -1 + 1 = 0
0 + x = 1.224744871 + 1
x = 1.224744871 + 1

Combine like terms: 1.224744871 + 1 = 2.224744871
x = 2.224744871

Simplifying
x = 2.224744871

Subproblem 2
x + -1 = -1.224744871

Simplifying
x + -1 = -1.224744871

Reorder the terms:
-1 + x = -1.224744871

Solving
-1 + x = -1.224744871

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 = -1.224744871 + 1

Combine like terms: -1 + 1 = 0
0 + x = -1.224744871 + 1
x = -1.224744871 + 1

Combine like terms: -1.224744871 + 1 = -0.224744871
x = -0.224744871

Simplifying
x = -0.224744871

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.224744871, -0.224744871}
Solution
x = {2.224744871, -0.224744871}