6x^2+8x-84=0

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Solution for 6x^2+8x-84=0 equation: 


Simplifying
6x2 + 8x + -84 = 0

Reorder the terms:
-84 + 8x + 6x2 = 0

Solving
-84 + 8x + 6x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '2'.
2(-42 + 4x + 3x2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-42 + 4x + 3x2)' equal to zero and attempt to solve:

Simplifying
-42 + 4x + 3x2 = 0

Solving
-42 + 4x + 3x2 = 0

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

Divide each side by '3'.
-14 + 1.333333333x + x2 = 0

Move the constant term to the right:

Add '14' to each side of the equation.
-14 + 1.333333333x + 14 + x2 = 0 + 14

Reorder the terms:
-14 + 14 + 1.333333333x + x2 = 0 + 14

Combine like terms: -14 + 14 = 0
0 + 1.333333333x + x2 = 0 + 14
1.333333333x + x2 = 0 + 14

Combine like terms: 0 + 14 = 14
1.333333333x + x2 = 14

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 = 14 + 0.4444444442

Reorder the terms:
0.4444444442 + 1.333333333x + x2 = 14 + 0.4444444442

Combine like terms: 14 + 0.4444444442 = 14.4444444442
0.4444444442 + 1.333333333x + x2 = 14.4444444442

Factor a perfect square on the left side:
(x + 0.6666666665)(x + 0.6666666665) = 14.4444444442

Calculate the square root of the right side: 3.80058475

Break this problem into two subproblems by setting 
(x + 0.6666666665) equal to 3.80058475 and -3.80058475.


Subproblem 1
x + 0.6666666665 = 3.80058475

Simplifying
x + 0.6666666665 = 3.80058475

Reorder the terms:
0.6666666665 + x = 3.80058475

Solving
0.6666666665 + x = 3.80058475

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 = 3.80058475 + -0.6666666665

Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000
0.0000000000 + x = 3.80058475 + -0.6666666665
x = 3.80058475 + -0.6666666665

Combine like terms: 3.80058475 + -0.6666666665 = 3.1339180835
x = 3.1339180835

Simplifying
x = 3.1339180835


Subproblem 2
x + 0.6666666665 = -3.80058475

Simplifying
x + 0.6666666665 = -3.80058475

Reorder the terms:
0.6666666665 + x = -3.80058475

Solving
0.6666666665 + x = -3.80058475

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 = -3.80058475 + -0.6666666665

Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000
0.0000000000 + x = -3.80058475 + -0.6666666665
x = -3.80058475 + -0.6666666665

Combine like terms: -3.80058475 + -0.6666666665 = -4.4672514165
x = -4.4672514165

Simplifying
x = -4.4672514165


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {3.1339180835, -4.4672514165}
Solution
x = {3.1339180835, -4.4672514165}

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