6x^2+x+1=0

Solution for 6x^2+x+1=0 equation:

Simplifying
6x2 + x + 1 = 0

Reorder the terms:
1 + x + 6x2 = 0

Solving
1 + x + 6x2 = 0

Solving for variable 'x'.

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

Divide each side by '6'.
0.1666666667 + 0.1666666667x + x2 = 0

Move the constant term to the right:

Add '-0.1666666667' to each side of the equation.
0.1666666667 + 0.1666666667x + -0.1666666667 + x2 = 0 + -0.1666666667

Reorder the terms:
0.1666666667 + -0.1666666667 + 0.1666666667x + x2 = 0 + -0.1666666667

Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000
0.0000000000 + 0.1666666667x + x2 = 0 + -0.1666666667
0.1666666667x + x2 = 0 + -0.1666666667

Combine like terms: 0 + -0.1666666667 = -0.1666666667
0.1666666667x + x2 = -0.1666666667

The x term is x.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
0.1666666667x + 0.25 + x2 = -0.1666666667 + 0.25

Reorder the terms:
0.25 + 0.1666666667x + x2 = -0.1666666667 + 0.25

Combine like terms: -0.1666666667 + 0.25 = 0.0833333333
0.25 + 0.1666666667x + x2 = 0.0833333333

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 0.0833333333

Calculate the square root of the right side: 0.288675135

Break this problem into two subproblems by setting 
(x + 0.5) equal to 0.288675135 and -0.288675135.

Subproblem 1
x + 0.5 = 0.288675135

Simplifying
x + 0.5 = 0.288675135

Reorder the terms:
0.5 + x = 0.288675135

Solving
0.5 + x = 0.288675135

Solving for variable 'x'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x = 0.288675135 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 0.288675135 + -0.5
x = 0.288675135 + -0.5

Combine like terms: 0.288675135 + -0.5 = -0.211324865
x = -0.211324865

Simplifying
x = -0.211324865

Subproblem 2
x + 0.5 = -0.288675135

Simplifying
x + 0.5 = -0.288675135

Reorder the terms:
0.5 + x = -0.288675135

Solving
0.5 + x = -0.288675135

Solving for variable 'x'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x = -0.288675135 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -0.288675135 + -0.5
x = -0.288675135 + -0.5

Combine like terms: -0.288675135 + -0.5 = -0.788675135
x = -0.788675135

Simplifying
x = -0.788675135

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.211324865, -0.788675135}