12x^2-32x-48=0

Solution for 12x^2-32x-48=0 equation:

Simplifying
12x2 + -32x + -48 = 0

Reorder the terms:
-48 + -32x + 12x2 = 0

Solving
-48 + -32x + 12x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '4'.
4(-12 + -8x + 3x2) = 0

Ignore the factor 4.
Subproblem 1
Set the factor '(-12 + -8x + 3x2)' equal to zero and attempt to solve:

Simplifying
-12 + -8x + 3x2 = 0

Solving
-12 + -8x + 3x2 = 0

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

Divide each side by '3'.
-4 + -2.666666667x + x2 = 0

Move the constant term to the right:

Add '4' to each side of the equation.
-4 + -2.666666667x + 4 + x2 = 0 + 4

Reorder the terms:
-4 + 4 + -2.666666667x + x2 = 0 + 4

Combine like terms: -4 + 4 = 0
0 + -2.666666667x + x2 = 0 + 4
-2.666666667x + x2 = 0 + 4

Combine like terms: 0 + 4 = 4
-2.666666667x + x2 = 4

The x term is -2.666666667x.  Take half its coefficient (-1.333333334).
Square it (1.777777780) and add it to both sides.

Add '1.777777780' to each side of the equation.
-2.666666667x + 1.777777780 + x2 = 4 + 1.777777780

Reorder the terms:
1.777777780 + -2.666666667x + x2 = 4 + 1.777777780

Combine like terms: 4 + 1.777777780 = 5.77777778
1.777777780 + -2.666666667x + x2 = 5.77777778

Factor a perfect square on the left side:
(x + -1.333333334)(x + -1.333333334) = 5.77777778

Calculate the square root of the right side: 2.403700851

Break this problem into two subproblems by setting 
(x + -1.333333334) equal to 2.403700851 and -2.403700851.

Subproblem 1
x + -1.333333334 = 2.403700851

Simplifying
x + -1.333333334 = 2.403700851

Reorder the terms:
-1.333333334 + x = 2.403700851

Solving
-1.333333334 + x = 2.403700851

Solving for variable 'x'.

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

Add '1.333333334' to each side of the equation.
-1.333333334 + 1.333333334 + x = 2.403700851 + 1.333333334

Combine like terms: -1.333333334 + 1.333333334 = 0.000000000
0.000000000 + x = 2.403700851 + 1.333333334
x = 2.403700851 + 1.333333334

Combine like terms: 2.403700851 + 1.333333334 = 3.737034185
x = 3.737034185

Simplifying
x = 3.737034185

Subproblem 2
x + -1.333333334 = -2.403700851

Simplifying
x + -1.333333334 = -2.403700851

Reorder the terms:
-1.333333334 + x = -2.403700851

Solving
-1.333333334 + x = -2.403700851

Solving for variable 'x'.

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

Add '1.333333334' to each side of the equation.
-1.333333334 + 1.333333334 + x = -2.403700851 + 1.333333334

Combine like terms: -1.333333334 + 1.333333334 = 0.000000000
0.000000000 + x = -2.403700851 + 1.333333334
x = -2.403700851 + 1.333333334

Combine like terms: -2.403700851 + 1.333333334 = -1.070367517
x = -1.070367517

Simplifying
x = -1.070367517

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {3.737034185, -1.070367517}
Solution
x = {3.737034185, -1.070367517}