12x^2-32x-117=0

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Solution for 12x^2-32x-117=0 equation: 


Simplifying
12x2 + -32x + -117 = 0

Reorder the terms:
-117 + -32x + 12x2 = 0

Solving
-117 + -32x + 12x2 = 0

Solving for variable 'x'.

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

Divide each side by '12'.
-9.75 + -2.666666667x + x2 = 0

Move the constant term to the right:

Add '9.75' to each side of the equation.
-9.75 + -2.666666667x + 9.75 + x2 = 0 + 9.75

Reorder the terms:
-9.75 + 9.75 + -2.666666667x + x2 = 0 + 9.75

Combine like terms: -9.75 + 9.75 = 0.00
0.00 + -2.666666667x + x2 = 0 + 9.75
-2.666666667x + x2 = 0 + 9.75

Combine like terms: 0 + 9.75 = 9.75
-2.666666667x + x2 = 9.75

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 = 9.75 + 1.777777780

Reorder the terms:
1.777777780 + -2.666666667x + x2 = 9.75 + 1.777777780

Combine like terms: 9.75 + 1.777777780 = 11.52777778
1.777777780 + -2.666666667x + x2 = 11.52777778

Factor a perfect square on the left side:
(x + -1.333333334)(x + -1.333333334) = 11.52777778

Calculate the square root of the right side: 3.395258132

Break this problem into two subproblems by setting 
(x + -1.333333334) equal to 3.395258132 and -3.395258132.


Subproblem 1
x + -1.333333334 = 3.395258132

Simplifying
x + -1.333333334 = 3.395258132

Reorder the terms:
-1.333333334 + x = 3.395258132

Solving
-1.333333334 + x = 3.395258132

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 = 3.395258132 + 1.333333334

Combine like terms: -1.333333334 + 1.333333334 = 0.000000000
0.000000000 + x = 3.395258132 + 1.333333334
x = 3.395258132 + 1.333333334

Combine like terms: 3.395258132 + 1.333333334 = 4.728591466
x = 4.728591466

Simplifying
x = 4.728591466


Subproblem 2
x + -1.333333334 = -3.395258132

Simplifying
x + -1.333333334 = -3.395258132

Reorder the terms:
-1.333333334 + x = -3.395258132

Solving
-1.333333334 + x = -3.395258132

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 = -3.395258132 + 1.333333334

Combine like terms: -1.333333334 + 1.333333334 = 0.000000000
0.000000000 + x = -3.395258132 + 1.333333334
x = -3.395258132 + 1.333333334

Combine like terms: -3.395258132 + 1.333333334 = -2.061924798
x = -2.061924798

Simplifying
x = -2.061924798


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {4.728591466, -2.061924798}

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