6x^2+31x-12=0

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Solution for 6x^2+31x-12=0 equation: 


Simplifying
6x2 + 31x + -12 = 0

Reorder the terms:
-12 + 31x + 6x2 = 0

Solving
-12 + 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'.
-2 + 5.166666667x + x2 = 0

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + 5.166666667x + 2 + x2 = 0 + 2

Reorder the terms:
-2 + 2 + 5.166666667x + x2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + 5.166666667x + x2 = 0 + 2
5.166666667x + x2 = 0 + 2

Combine like terms: 0 + 2 = 2
5.166666667x + x2 = 2

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 = 2 + 6.673611115

Reorder the terms:
6.673611115 + 5.166666667x + x2 = 2 + 6.673611115

Combine like terms: 2 + 6.673611115 = 8.673611115
6.673611115 + 5.166666667x + x2 = 8.673611115

Factor a perfect square on the left side:
(x + 2.583333334)(x + 2.583333334) = 8.673611115

Calculate the square root of the right side: 2.945099509

Break this problem into two subproblems by setting 
(x + 2.583333334) equal to 2.945099509 and -2.945099509.


Subproblem 1
x + 2.583333334 = 2.945099509

Simplifying
x + 2.583333334 = 2.945099509

Reorder the terms:
2.583333334 + x = 2.945099509

Solving
2.583333334 + x = 2.945099509

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 = 2.945099509 + -2.583333334

Combine like terms: 2.583333334 + -2.583333334 = 0.000000000
0.000000000 + x = 2.945099509 + -2.583333334
x = 2.945099509 + -2.583333334

Combine like terms: 2.945099509 + -2.583333334 = 0.361766175
x = 0.361766175

Simplifying
x = 0.361766175


Subproblem 2
x + 2.583333334 = -2.945099509

Simplifying
x + 2.583333334 = -2.945099509

Reorder the terms:
2.583333334 + x = -2.945099509

Solving
2.583333334 + x = -2.945099509

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 = -2.945099509 + -2.583333334

Combine like terms: 2.583333334 + -2.583333334 = 0.000000000
0.000000000 + x = -2.945099509 + -2.583333334
x = -2.945099509 + -2.583333334

Combine like terms: -2.945099509 + -2.583333334 = -5.528432843
x = -5.528432843

Simplifying
x = -5.528432843


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.361766175, -5.528432843}

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