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 = 4.673611115
6.673611115 + -5.166666667x + x2 = 4.673611115

Factor a perfect square on the left side:
(x + -2.583333334)(x + -2.583333334) = 4.673611115

Calculate the square root of the right side: 2.161853629

Break this problem into two subproblems by setting 
(x + -2.583333334) equal to 2.161853629 and -2.161853629.


Subproblem 1
x + -2.583333334 = 2.161853629

Simplifying
x + -2.583333334 = 2.161853629

Reorder the terms:
-2.583333334 + x = 2.161853629

Solving
-2.583333334 + x = 2.161853629

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 = 2.161853629 + 2.583333334

Combine like terms: -2.583333334 + 2.583333334 = 0.000000000
0.000000000 + x = 2.161853629 + 2.583333334
x = 2.161853629 + 2.583333334

Combine like terms: 2.161853629 + 2.583333334 = 4.745186963
x = 4.745186963

Simplifying
x = 4.745186963


Subproblem 2
x + -2.583333334 = -2.161853629

Simplifying
x + -2.583333334 = -2.161853629

Reorder the terms:
-2.583333334 + x = -2.161853629

Solving
-2.583333334 + x = -2.161853629

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 = -2.161853629 + 2.583333334

Combine like terms: -2.583333334 + 2.583333334 = 0.000000000
0.000000000 + x = -2.161853629 + 2.583333334
x = -2.161853629 + 2.583333334

Combine like terms: -2.161853629 + 2.583333334 = 0.421479705
x = 0.421479705

Simplifying
x = 0.421479705


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {4.745186963, 0.421479705}

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