3x^2-20x+31=0

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Solution for 3x^2-20x+31=0 equation: 


Simplifying
3x2 + -20x + 31 = 0

Reorder the terms:
31 + -20x + 3x2 = 0

Solving
31 + -20x + 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 + -6.666666667x + x2 = 0

Move the constant term to the right:

Add '-10.33333333' to each side of the equation.
10.33333333 + -6.666666667x + -10.33333333 + x2 = 0 + -10.33333333

Reorder the terms:
10.33333333 + -10.33333333 + -6.666666667x + x2 = 0 + -10.33333333

Combine like terms: 10.33333333 + -10.33333333 = 0.00000000
0.00000000 + -6.666666667x + x2 = 0 + -10.33333333
-6.666666667x + x2 = 0 + -10.33333333

Combine like terms: 0 + -10.33333333 = -10.33333333
-6.666666667x + x2 = -10.33333333

The x term is -6.666666667x.  Take half its coefficient (-3.333333334).
Square it (11.11111112) and add it to both sides.

Add '11.11111112' to each side of the equation.
-6.666666667x + 11.11111112 + x2 = -10.33333333 + 11.11111112

Reorder the terms:
11.11111112 + -6.666666667x + x2 = -10.33333333 + 11.11111112

Combine like terms: -10.33333333 + 11.11111112 = 0.77777779
11.11111112 + -6.666666667x + x2 = 0.77777779

Factor a perfect square on the left side:
(x + -3.333333334)(x + -3.333333334) = 0.77777779

Calculate the square root of the right side: 0.881917111

Break this problem into two subproblems by setting 
(x + -3.333333334) equal to 0.881917111 and -0.881917111.


Subproblem 1
x + -3.333333334 = 0.881917111

Simplifying
x + -3.333333334 = 0.881917111

Reorder the terms:
-3.333333334 + x = 0.881917111

Solving
-3.333333334 + x = 0.881917111

Solving for variable 'x'.

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

Add '3.333333334' to each side of the equation.
-3.333333334 + 3.333333334 + x = 0.881917111 + 3.333333334

Combine like terms: -3.333333334 + 3.333333334 = 0.000000000
0.000000000 + x = 0.881917111 + 3.333333334
x = 0.881917111 + 3.333333334

Combine like terms: 0.881917111 + 3.333333334 = 4.215250445
x = 4.215250445

Simplifying
x = 4.215250445


Subproblem 2
x + -3.333333334 = -0.881917111

Simplifying
x + -3.333333334 = -0.881917111

Reorder the terms:
-3.333333334 + x = -0.881917111

Solving
-3.333333334 + x = -0.881917111

Solving for variable 'x'.

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

Add '3.333333334' to each side of the equation.
-3.333333334 + 3.333333334 + x = -0.881917111 + 3.333333334

Combine like terms: -3.333333334 + 3.333333334 = 0.000000000
0.000000000 + x = -0.881917111 + 3.333333334
x = -0.881917111 + 3.333333334

Combine like terms: -0.881917111 + 3.333333334 = 2.451416223
x = 2.451416223

Simplifying
x = 2.451416223


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {4.215250445, 2.451416223}

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