3x^2-31x+13=0

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Solution for 3x^2-31x+13=0 equation: 


Simplifying
3x2 + -31x + 13 = 0

Reorder the terms:
13 + -31x + 3x2 = 0

Solving
13 + -31x + 3x2 = 0

Solving for variable 'x'.

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

Divide each side by '3'.
4.333333333 + -10.33333333x + x2 = 0

Move the constant term to the right:

Add '-4.333333333' to each side of the equation.
4.333333333 + -10.33333333x + -4.333333333 + x2 = 0 + -4.333333333

Reorder the terms:
4.333333333 + -4.333333333 + -10.33333333x + x2 = 0 + -4.333333333

Combine like terms: 4.333333333 + -4.333333333 = 0.000000000
0.000000000 + -10.33333333x + x2 = 0 + -4.333333333
-10.33333333x + x2 = 0 + -4.333333333

Combine like terms: 0 + -4.333333333 = -4.333333333
-10.33333333x + x2 = -4.333333333

The x term is -10.33333333x.  Take half its coefficient (-5.166666665).
Square it (26.69444443) and add it to both sides.

Add '26.69444443' to each side of the equation.
-10.33333333x + 26.69444443 + x2 = -4.333333333 + 26.69444443

Reorder the terms:
26.69444443 + -10.33333333x + x2 = -4.333333333 + 26.69444443

Combine like terms: -4.333333333 + 26.69444443 = 22.361111097
26.69444443 + -10.33333333x + x2 = 22.361111097

Factor a perfect square on the left side:
(x + -5.166666665)(x + -5.166666665) = 22.361111097

Calculate the square root of the right side: 4.728753652

Break this problem into two subproblems by setting 
(x + -5.166666665) equal to 4.728753652 and -4.728753652.


Subproblem 1
x + -5.166666665 = 4.728753652

Simplifying
x + -5.166666665 = 4.728753652

Reorder the terms:
-5.166666665 + x = 4.728753652

Solving
-5.166666665 + x = 4.728753652

Solving for variable 'x'.

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

Add '5.166666665' to each side of the equation.
-5.166666665 + 5.166666665 + x = 4.728753652 + 5.166666665

Combine like terms: -5.166666665 + 5.166666665 = 0.000000000
0.000000000 + x = 4.728753652 + 5.166666665
x = 4.728753652 + 5.166666665

Combine like terms: 4.728753652 + 5.166666665 = 9.895420317
x = 9.895420317

Simplifying
x = 9.895420317


Subproblem 2
x + -5.166666665 = -4.728753652

Simplifying
x + -5.166666665 = -4.728753652

Reorder the terms:
-5.166666665 + x = -4.728753652

Solving
-5.166666665 + x = -4.728753652

Solving for variable 'x'.

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

Add '5.166666665' to each side of the equation.
-5.166666665 + 5.166666665 + x = -4.728753652 + 5.166666665

Combine like terms: -5.166666665 + 5.166666665 = 0.000000000
0.000000000 + x = -4.728753652 + 5.166666665
x = -4.728753652 + 5.166666665

Combine like terms: -4.728753652 + 5.166666665 = 0.437913013
x = 0.437913013

Simplifying
x = 0.437913013


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {9.895420317, 0.437913013}

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