6x=3x^2+1

Solution for 6x=3x^2+1 equation:

Simplifying
6x = 3x2 + 1

Reorder the terms:
6x = 1 + 3x2

Solving
6x = 1 + 3x2

Solving for variable 'x'.

Reorder the terms:
-1 + 6x + -3x2 = 1 + 3x2 + -1 + -3x2

Reorder the terms:
-1 + 6x + -3x2 = 1 + -1 + 3x2 + -3x2

Combine like terms: 1 + -1 = 0
-1 + 6x + -3x2 = 0 + 3x2 + -3x2
-1 + 6x + -3x2 = 3x2 + -3x2

Combine like terms: 3x2 + -3x2 = 0
-1 + 6x + -3x2 = 0

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

Divide each side by '-3'.
0.3333333333 + -2x + x2 = 0

Move the constant term to the right:

Add '-0.3333333333' to each side of the equation.
0.3333333333 + -2x + -0.3333333333 + x2 = 0 + -0.3333333333

Reorder the terms:
0.3333333333 + -0.3333333333 + -2x + x2 = 0 + -0.3333333333

Combine like terms: 0.3333333333 + -0.3333333333 = 0.0000000000
0.0000000000 + -2x + x2 = 0 + -0.3333333333
-2x + x2 = 0 + -0.3333333333

Combine like terms: 0 + -0.3333333333 = -0.3333333333
-2x + x2 = -0.3333333333

The x term is -2x.  Take half its coefficient (-1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
-2x + 1 + x2 = -0.3333333333 + 1

Reorder the terms:
1 + -2x + x2 = -0.3333333333 + 1

Combine like terms: -0.3333333333 + 1 = 0.6666666667
1 + -2x + x2 = 0.6666666667

Factor a perfect square on the left side:
(x + -1)(x + -1) = 0.6666666667

Calculate the square root of the right side: 0.816496581

Break this problem into two subproblems by setting 
(x + -1) equal to 0.816496581 and -0.816496581.

Subproblem 1
x + -1 = 0.816496581

Simplifying
x + -1 = 0.816496581

Reorder the terms:
-1 + x = 0.816496581

Solving
-1 + x = 0.816496581

Solving for variable 'x'.

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

Add '1' to each side of the equation.
-1 + 1 + x = 0.816496581 + 1

Combine like terms: -1 + 1 = 0
0 + x = 0.816496581 + 1
x = 0.816496581 + 1

Combine like terms: 0.816496581 + 1 = 1.816496581
x = 1.816496581

Simplifying
x = 1.816496581

Subproblem 2
x + -1 = -0.816496581

Simplifying
x + -1 = -0.816496581

Reorder the terms:
-1 + x = -0.816496581

Solving
-1 + x = -0.816496581

Solving for variable 'x'.

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

Add '1' to each side of the equation.
-1 + 1 + x = -0.816496581 + 1

Combine like terms: -1 + 1 = 0
0 + x = -0.816496581 + 1
x = -0.816496581 + 1

Combine like terms: -0.816496581 + 1 = 0.183503419
x = 0.183503419

Simplifying
x = 0.183503419

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.816496581, 0.183503419}