12x^2-24x-27=0

Solution for 12x^2-24x-27=0 equation:

Simplifying
12x2 + -24x + -27 = 0

Reorder the terms:
-27 + -24x + 12x2 = 0

Solving
-27 + -24x + 12x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '3'.
3(-9 + -8x + 4x2) = 0

Ignore the factor 3.
Subproblem 1
Set the factor '(-9 + -8x + 4x2)' equal to zero and attempt to solve:

Simplifying
-9 + -8x + 4x2 = 0

Solving
-9 + -8x + 4x2 = 0

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

Divide each side by '4'.
-2.25 + -2x + x2 = 0

Move the constant term to the right:

Add '2.25' to each side of the equation.
-2.25 + -2x + 2.25 + x2 = 0 + 2.25

Reorder the terms:
-2.25 + 2.25 + -2x + x2 = 0 + 2.25

Combine like terms: -2.25 + 2.25 = 0.00
0.00 + -2x + x2 = 0 + 2.25
-2x + x2 = 0 + 2.25

Combine like terms: 0 + 2.25 = 2.25
-2x + x2 = 2.25

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 = 2.25 + 1

Reorder the terms:
1 + -2x + x2 = 2.25 + 1

Combine like terms: 2.25 + 1 = 3.25
1 + -2x + x2 = 3.25

Factor a perfect square on the left side:
(x + -1)(x + -1) = 3.25

Calculate the square root of the right side: 1.802775638

Break this problem into two subproblems by setting 
(x + -1) equal to 1.802775638 and -1.802775638.

Subproblem 1
x + -1 = 1.802775638

Simplifying
x + -1 = 1.802775638

Reorder the terms:
-1 + x = 1.802775638

Solving
-1 + x = 1.802775638

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 = 1.802775638 + 1

Combine like terms: -1 + 1 = 0
0 + x = 1.802775638 + 1
x = 1.802775638 + 1

Combine like terms: 1.802775638 + 1 = 2.802775638
x = 2.802775638

Simplifying
x = 2.802775638

Subproblem 2
x + -1 = -1.802775638

Simplifying
x + -1 = -1.802775638

Reorder the terms:
-1 + x = -1.802775638

Solving
-1 + x = -1.802775638

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 = -1.802775638 + 1

Combine like terms: -1 + 1 = 0
0 + x = -1.802775638 + 1
x = -1.802775638 + 1

Combine like terms: -1.802775638 + 1 = -0.802775638
x = -0.802775638

Simplifying
x = -0.802775638

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.802775638, -0.802775638}
Solution
x = {2.802775638, -0.802775638}