6x^2+8x-83=0

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Solution for 6x^2+8x-83=0 equation: 


Simplifying
6x2 + 8x + -83 = 0

Reorder the terms:
-83 + 8x + 6x2 = 0

Solving
-83 + 8x + 6x2 = 0

Solving for variable 'x'.

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

Divide each side by '6'.
-13.83333333 + 1.333333333x + x2 = 0

Move the constant term to the right:

Add '13.83333333' to each side of the equation.
-13.83333333 + 1.333333333x + 13.83333333 + x2 = 0 + 13.83333333

Reorder the terms:
-13.83333333 + 13.83333333 + 1.333333333x + x2 = 0 + 13.83333333

Combine like terms: -13.83333333 + 13.83333333 = 0.00000000
0.00000000 + 1.333333333x + x2 = 0 + 13.83333333
1.333333333x + x2 = 0 + 13.83333333

Combine like terms: 0 + 13.83333333 = 13.83333333
1.333333333x + x2 = 13.83333333

The x term is 1.333333333x.  Take half its coefficient (0.6666666665).
Square it (0.4444444442) and add it to both sides.

Add '0.4444444442' to each side of the equation.
1.333333333x + 0.4444444442 + x2 = 13.83333333 + 0.4444444442

Reorder the terms:
0.4444444442 + 1.333333333x + x2 = 13.83333333 + 0.4444444442

Combine like terms: 13.83333333 + 0.4444444442 = 14.2777777742
0.4444444442 + 1.333333333x + x2 = 14.2777777742

Factor a perfect square on the left side:
(x + 0.6666666665)(x + 0.6666666665) = 14.2777777742

Calculate the square root of the right side: 3.778594682

Break this problem into two subproblems by setting 
(x + 0.6666666665) equal to 3.778594682 and -3.778594682.


Subproblem 1
x + 0.6666666665 = 3.778594682

Simplifying
x + 0.6666666665 = 3.778594682

Reorder the terms:
0.6666666665 + x = 3.778594682

Solving
0.6666666665 + x = 3.778594682

Solving for variable 'x'.

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

Add '-0.6666666665' to each side of the equation.
0.6666666665 + -0.6666666665 + x = 3.778594682 + -0.6666666665

Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000
0.0000000000 + x = 3.778594682 + -0.6666666665
x = 3.778594682 + -0.6666666665

Combine like terms: 3.778594682 + -0.6666666665 = 3.1119280155
x = 3.1119280155

Simplifying
x = 3.1119280155


Subproblem 2
x + 0.6666666665 = -3.778594682

Simplifying
x + 0.6666666665 = -3.778594682

Reorder the terms:
0.6666666665 + x = -3.778594682

Solving
0.6666666665 + x = -3.778594682

Solving for variable 'x'.

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

Add '-0.6666666665' to each side of the equation.
0.6666666665 + -0.6666666665 + x = -3.778594682 + -0.6666666665

Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000
0.0000000000 + x = -3.778594682 + -0.6666666665
x = -3.778594682 + -0.6666666665

Combine like terms: -3.778594682 + -0.6666666665 = -4.4452613485
x = -4.4452613485

Simplifying
x = -4.4452613485


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {3.1119280155, -4.4452613485}

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