9w^2+w+1=0

Solution for 9w^2+w+1=0 equation:

Simplifying
9w2 + w + 1 = 0

Reorder the terms:
1 + w + 9w2 = 0

Solving
1 + w + 9w2 = 0

Solving for variable 'w'.

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

Divide each side by '9'.
0.1111111111 + 0.1111111111w + w2 = 0

Move the constant term to the right:

Add '-0.1111111111' to each side of the equation.
0.1111111111 + 0.1111111111w + -0.1111111111 + w2 = 0 + -0.1111111111

Reorder the terms:
0.1111111111 + -0.1111111111 + 0.1111111111w + w2 = 0 + -0.1111111111

Combine like terms: 0.1111111111 + -0.1111111111 = 0.0000000000
0.0000000000 + 0.1111111111w + w2 = 0 + -0.1111111111
0.1111111111w + w2 = 0 + -0.1111111111

Combine like terms: 0 + -0.1111111111 = -0.1111111111
0.1111111111w + w2 = -0.1111111111

The w term is w.  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.
0.1111111111w + 0.25 + w2 = -0.1111111111 + 0.25

Reorder the terms:
0.25 + 0.1111111111w + w2 = -0.1111111111 + 0.25

Combine like terms: -0.1111111111 + 0.25 = 0.1388888889
0.25 + 0.1111111111w + w2 = 0.1388888889

Factor a perfect square on the left side:
(w + 0.5)(w + 0.5) = 0.1388888889

Calculate the square root of the right side: 0.372677996

Break this problem into two subproblems by setting 
(w + 0.5) equal to 0.372677996 and -0.372677996.

Subproblem 1
w + 0.5 = 0.372677996

Simplifying
w + 0.5 = 0.372677996

Reorder the terms:
0.5 + w = 0.372677996

Solving
0.5 + w = 0.372677996

Solving for variable 'w'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + w = 0.372677996 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + w = 0.372677996 + -0.5
w = 0.372677996 + -0.5

Combine like terms: 0.372677996 + -0.5 = -0.127322004
w = -0.127322004

Simplifying
w = -0.127322004

Subproblem 2
w + 0.5 = -0.372677996

Simplifying
w + 0.5 = -0.372677996

Reorder the terms:
0.5 + w = -0.372677996

Solving
0.5 + w = -0.372677996

Solving for variable 'w'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + w = -0.372677996 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + w = -0.372677996 + -0.5
w = -0.372677996 + -0.5

Combine like terms: -0.372677996 + -0.5 = -0.872677996
w = -0.872677996

Simplifying
w = -0.872677996

Solution
The solution to the problem is based on the solutions
from the subproblems.
w = {-0.127322004, -0.872677996}