3x^2-37x+31=0

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Solution for 3x^2-37x+31=0 equation:


Simplifying
3x2 + -37x + 31 = 0

Reorder the terms:
31 + -37x + 3x2 = 0

Solving
31 + -37x + 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'.
10.33333333 + -12.33333333x + x2 = 0

Move the constant term to the right:

Add '-10.33333333' to each side of the equation.
10.33333333 + -12.33333333x + -10.33333333 + x2 = 0 + -10.33333333

Reorder the terms:
10.33333333 + -10.33333333 + -12.33333333x + x2 = 0 + -10.33333333

Combine like terms: 10.33333333 + -10.33333333 = 0.00000000
0.00000000 + -12.33333333x + x2 = 0 + -10.33333333
-12.33333333x + x2 = 0 + -10.33333333

Combine like terms: 0 + -10.33333333 = -10.33333333
-12.33333333x + x2 = -10.33333333

The x term is -12.33333333x.  Take half its coefficient (-6.166666665).
Square it (38.02777776) and add it to both sides.

Add '38.02777776' to each side of the equation.
-12.33333333x + 38.02777776 + x2 = -10.33333333 + 38.02777776

Reorder the terms:
38.02777776 + -12.33333333x + x2 = -10.33333333 + 38.02777776

Combine like terms: -10.33333333 + 38.02777776 = 27.69444443
38.02777776 + -12.33333333x + x2 = 27.69444443

Factor a perfect square on the left side:
(x + -6.166666665)(x + -6.166666665) = 27.69444443

Calculate the square root of the right side: 5.262551133

Break this problem into two subproblems by setting 
(x + -6.166666665) equal to 5.262551133 and -5.262551133.


Subproblem 1
x + -6.166666665 = 5.262551133

Simplifying
x + -6.166666665 = 5.262551133

Reorder the terms:
-6.166666665 + x = 5.262551133

Solving
-6.166666665 + x = 5.262551133

Solving for variable 'x'.

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

Add '6.166666665' to each side of the equation.
-6.166666665 + 6.166666665 + x = 5.262551133 + 6.166666665

Combine like terms: -6.166666665 + 6.166666665 = 0.000000000
0.000000000 + x = 5.262551133 + 6.166666665
x = 5.262551133 + 6.166666665

Combine like terms: 5.262551133 + 6.166666665 = 11.429217798
x = 11.429217798

Simplifying
x = 11.429217798


Subproblem 2
x + -6.166666665 = -5.262551133

Simplifying
x + -6.166666665 = -5.262551133

Reorder the terms:
-6.166666665 + x = -5.262551133

Solving
-6.166666665 + x = -5.262551133

Solving for variable 'x'.

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

Add '6.166666665' to each side of the equation.
-6.166666665 + 6.166666665 + x = -5.262551133 + 6.166666665

Combine like terms: -6.166666665 + 6.166666665 = 0.000000000
0.000000000 + x = -5.262551133 + 6.166666665
x = -5.262551133 + 6.166666665

Combine like terms: -5.262551133 + 6.166666665 = 0.904115532
x = 0.904115532

Simplifying
x = 0.904115532


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {11.429217798, 0.904115532}

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