3x^2-40x=-31

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Solution for 3x^2-40x=-31 equation: 


Simplifying
3x2 + -40x = -31

Reorder the terms:
-40x + 3x2 = -31

Solving
-40x + 3x2 = -31

Solving for variable 'x'.

Reorder the terms:
31 + -40x + 3x2 = -31 + 31

Combine like terms: -31 + 31 = 0
31 + -40x + 3x2 = 0

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

Divide each side by '3'.
10.33333333 + -13.33333333x + x2 = 0

Move the constant term to the right:

Add '-10.33333333' to each side of the equation.
10.33333333 + -13.33333333x + -10.33333333 + x2 = 0 + -10.33333333

Reorder the terms:
10.33333333 + -10.33333333 + -13.33333333x + x2 = 0 + -10.33333333

Combine like terms: 10.33333333 + -10.33333333 = 0.00000000
0.00000000 + -13.33333333x + x2 = 0 + -10.33333333
-13.33333333x + x2 = 0 + -10.33333333

Combine like terms: 0 + -10.33333333 = -10.33333333
-13.33333333x + x2 = -10.33333333

The x term is -13.33333333x.  Take half its coefficient (-6.666666665).
Square it (44.44444442) and add it to both sides.

Add '44.44444442' to each side of the equation.
-13.33333333x + 44.44444442 + x2 = -10.33333333 + 44.44444442

Reorder the terms:
44.44444442 + -13.33333333x + x2 = -10.33333333 + 44.44444442

Combine like terms: -10.33333333 + 44.44444442 = 34.11111109
44.44444442 + -13.33333333x + x2 = 34.11111109

Factor a perfect square on the left side:
(x + -6.666666665)(x + -6.666666665) = 34.11111109

Calculate the square root of the right side: 5.840471821

Break this problem into two subproblems by setting 
(x + -6.666666665) equal to 5.840471821 and -5.840471821.


Subproblem 1
x + -6.666666665 = 5.840471821

Simplifying
x + -6.666666665 = 5.840471821

Reorder the terms:
-6.666666665 + x = 5.840471821

Solving
-6.666666665 + x = 5.840471821

Solving for variable 'x'.

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

Add '6.666666665' to each side of the equation.
-6.666666665 + 6.666666665 + x = 5.840471821 + 6.666666665

Combine like terms: -6.666666665 + 6.666666665 = 0.000000000
0.000000000 + x = 5.840471821 + 6.666666665
x = 5.840471821 + 6.666666665

Combine like terms: 5.840471821 + 6.666666665 = 12.507138486
x = 12.507138486

Simplifying
x = 12.507138486


Subproblem 2
x + -6.666666665 = -5.840471821

Simplifying
x + -6.666666665 = -5.840471821

Reorder the terms:
-6.666666665 + x = -5.840471821

Solving
-6.666666665 + x = -5.840471821

Solving for variable 'x'.

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

Add '6.666666665' to each side of the equation.
-6.666666665 + 6.666666665 + x = -5.840471821 + 6.666666665

Combine like terms: -6.666666665 + 6.666666665 = 0.000000000
0.000000000 + x = -5.840471821 + 6.666666665
x = -5.840471821 + 6.666666665

Combine like terms: -5.840471821 + 6.666666665 = 0.826194844
x = 0.826194844

Simplifying
x = 0.826194844


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {12.507138486, 0.826194844}

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