-9w^2-5w+1=0

Solution for -9w^2-5w+1=0 equation:

Simplifying
-9w2 + -5w + 1 = 0

Reorder the terms:
1 + -5w + -9w2 = 0

Solving
1 + -5w + -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 + 0.5555555556w + w2 = 0

Move the constant term to the right:

Add '0.1111111111' to each side of the equation.
-0.1111111111 + 0.5555555556w + 0.1111111111 + w2 = 0 + 0.1111111111

Reorder the terms:
-0.1111111111 + 0.1111111111 + 0.5555555556w + w2 = 0 + 0.1111111111

Combine like terms: -0.1111111111 + 0.1111111111 = 0.0000000000
0.0000000000 + 0.5555555556w + w2 = 0 + 0.1111111111
0.5555555556w + w2 = 0 + 0.1111111111

Combine like terms: 0 + 0.1111111111 = 0.1111111111
0.5555555556w + w2 = 0.1111111111

The w term is 0.5555555556w.  Take half its coefficient (0.2777777778).
Square it (0.07716049384) and add it to both sides.

Add '0.07716049384' to each side of the equation.
0.5555555556w + 0.07716049384 + w2 = 0.1111111111 + 0.07716049384

Reorder the terms:
0.07716049384 + 0.5555555556w + w2 = 0.1111111111 + 0.07716049384

Combine like terms: 0.1111111111 + 0.07716049384 = 0.18827160494
0.07716049384 + 0.5555555556w + w2 = 0.18827160494

Factor a perfect square on the left side:
(w + 0.2777777778)(w + 0.2777777778) = 0.18827160494

Calculate the square root of the right side: 0.43390276

Break this problem into two subproblems by setting 
(w + 0.2777777778) equal to 0.43390276 and -0.43390276.

Subproblem 1
w + 0.2777777778 = 0.43390276

Simplifying
w + 0.2777777778 = 0.43390276

Reorder the terms:
0.2777777778 + w = 0.43390276

Solving
0.2777777778 + w = 0.43390276

Solving for variable 'w'.

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

Add '-0.2777777778' to each side of the equation.
0.2777777778 + -0.2777777778 + w = 0.43390276 + -0.2777777778

Combine like terms: 0.2777777778 + -0.2777777778 = 0.0000000000
0.0000000000 + w = 0.43390276 + -0.2777777778
w = 0.43390276 + -0.2777777778

Combine like terms: 0.43390276 + -0.2777777778 = 0.1561249822
w = 0.1561249822

Simplifying
w = 0.1561249822

Subproblem 2
w + 0.2777777778 = -0.43390276

Simplifying
w + 0.2777777778 = -0.43390276

Reorder the terms:
0.2777777778 + w = -0.43390276

Solving
0.2777777778 + w = -0.43390276

Solving for variable 'w'.

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

Add '-0.2777777778' to each side of the equation.
0.2777777778 + -0.2777777778 + w = -0.43390276 + -0.2777777778

Combine like terms: 0.2777777778 + -0.2777777778 = 0.0000000000
0.0000000000 + w = -0.43390276 + -0.2777777778
w = -0.43390276 + -0.2777777778

Combine like terms: -0.43390276 + -0.2777777778 = -0.7116805378
w = -0.7116805378

Simplifying
w = -0.7116805378

Solution
The solution to the problem is based on the solutions
from the subproblems.
w = {0.1561249822, -0.7116805378}