6x^2-31x-28=0

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Solution for 6x^2-31x-28=0 equation: 


Simplifying
6x2 + -31x + -28 = 0

Reorder the terms:
-28 + -31x + 6x2 = 0

Solving
-28 + -31x + 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'.
-4.666666667 + -5.166666667x + x2 = 0

Move the constant term to the right:

Add '4.666666667' to each side of the equation.
-4.666666667 + -5.166666667x + 4.666666667 + x2 = 0 + 4.666666667

Reorder the terms:
-4.666666667 + 4.666666667 + -5.166666667x + x2 = 0 + 4.666666667

Combine like terms: -4.666666667 + 4.666666667 = 0.000000000
0.000000000 + -5.166666667x + x2 = 0 + 4.666666667
-5.166666667x + x2 = 0 + 4.666666667

Combine like terms: 0 + 4.666666667 = 4.666666667
-5.166666667x + x2 = 4.666666667

The x term is -5.166666667x.  Take half its coefficient (-2.583333334).
Square it (6.673611115) and add it to both sides.

Add '6.673611115' to each side of the equation.
-5.166666667x + 6.673611115 + x2 = 4.666666667 + 6.673611115

Reorder the terms:
6.673611115 + -5.166666667x + x2 = 4.666666667 + 6.673611115

Combine like terms: 4.666666667 + 6.673611115 = 11.340277782
6.673611115 + -5.166666667x + x2 = 11.340277782

Factor a perfect square on the left side:
(x + -2.583333334)(x + -2.583333334) = 11.340277782

Calculate the square root of the right side: 3.367532892

Break this problem into two subproblems by setting 
(x + -2.583333334) equal to 3.367532892 and -3.367532892.


Subproblem 1
x + -2.583333334 = 3.367532892

Simplifying
x + -2.583333334 = 3.367532892

Reorder the terms:
-2.583333334 + x = 3.367532892

Solving
-2.583333334 + x = 3.367532892

Solving for variable 'x'.

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

Add '2.583333334' to each side of the equation.
-2.583333334 + 2.583333334 + x = 3.367532892 + 2.583333334

Combine like terms: -2.583333334 + 2.583333334 = 0.000000000
0.000000000 + x = 3.367532892 + 2.583333334
x = 3.367532892 + 2.583333334

Combine like terms: 3.367532892 + 2.583333334 = 5.950866226
x = 5.950866226

Simplifying
x = 5.950866226


Subproblem 2
x + -2.583333334 = -3.367532892

Simplifying
x + -2.583333334 = -3.367532892

Reorder the terms:
-2.583333334 + x = -3.367532892

Solving
-2.583333334 + x = -3.367532892

Solving for variable 'x'.

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

Add '2.583333334' to each side of the equation.
-2.583333334 + 2.583333334 + x = -3.367532892 + 2.583333334

Combine like terms: -2.583333334 + 2.583333334 = 0.000000000
0.000000000 + x = -3.367532892 + 2.583333334
x = -3.367532892 + 2.583333334

Combine like terms: -3.367532892 + 2.583333334 = -0.784199558
x = -0.784199558

Simplifying
x = -0.784199558


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {5.950866226, -0.784199558}

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