12x^2-13x-32=0

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Solution for 12x^2-13x-32=0 equation: 


Simplifying
12x2 + -13x + -32 = 0

Reorder the terms:
-32 + -13x + 12x2 = 0

Solving
-32 + -13x + 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'.
-2.666666667 + -1.083333333x + x2 = 0

Move the constant term to the right:

Add '2.666666667' to each side of the equation.
-2.666666667 + -1.083333333x + 2.666666667 + x2 = 0 + 2.666666667

Reorder the terms:
-2.666666667 + 2.666666667 + -1.083333333x + x2 = 0 + 2.666666667

Combine like terms: -2.666666667 + 2.666666667 = 0.000000000
0.000000000 + -1.083333333x + x2 = 0 + 2.666666667
-1.083333333x + x2 = 0 + 2.666666667

Combine like terms: 0 + 2.666666667 = 2.666666667
-1.083333333x + x2 = 2.666666667

The x term is -1.083333333x.  Take half its coefficient (-0.5416666665).
Square it (0.2934027776) and add it to both sides.

Add '0.2934027776' to each side of the equation.
-1.083333333x + 0.2934027776 + x2 = 2.666666667 + 0.2934027776

Reorder the terms:
0.2934027776 + -1.083333333x + x2 = 2.666666667 + 0.2934027776

Combine like terms: 2.666666667 + 0.2934027776 = 2.9600694446
0.2934027776 + -1.083333333x + x2 = 2.9600694446

Factor a perfect square on the left side:
(x + -0.5416666665)(x + -0.5416666665) = 2.9600694446

Calculate the square root of the right side: 1.720485235

Break this problem into two subproblems by setting 
(x + -0.5416666665) equal to 1.720485235 and -1.720485235.


Subproblem 1
x + -0.5416666665 = 1.720485235

Simplifying
x + -0.5416666665 = 1.720485235

Reorder the terms:
-0.5416666665 + x = 1.720485235

Solving
-0.5416666665 + x = 1.720485235

Solving for variable 'x'.

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

Add '0.5416666665' to each side of the equation.
-0.5416666665 + 0.5416666665 + x = 1.720485235 + 0.5416666665

Combine like terms: -0.5416666665 + 0.5416666665 = 0.0000000000
0.0000000000 + x = 1.720485235 + 0.5416666665
x = 1.720485235 + 0.5416666665

Combine like terms: 1.720485235 + 0.5416666665 = 2.2621519015
x = 2.2621519015

Simplifying
x = 2.2621519015


Subproblem 2
x + -0.5416666665 = -1.720485235

Simplifying
x + -0.5416666665 = -1.720485235

Reorder the terms:
-0.5416666665 + x = -1.720485235

Solving
-0.5416666665 + x = -1.720485235

Solving for variable 'x'.

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

Add '0.5416666665' to each side of the equation.
-0.5416666665 + 0.5416666665 + x = -1.720485235 + 0.5416666665

Combine like terms: -0.5416666665 + 0.5416666665 = 0.0000000000
0.0000000000 + x = -1.720485235 + 0.5416666665
x = -1.720485235 + 0.5416666665

Combine like terms: -1.720485235 + 0.5416666665 = -1.1788185685
x = -1.1788185685

Simplifying
x = -1.1788185685


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.2621519015, -1.1788185685}

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