9w^2-4w-1=0

Solution for 9w^2-4w-1=0 equation:

Simplifying
9w2 + -4w + -1 = 0

Reorder the terms:
-1 + -4w + 9w2 = 0

Solving
-1 + -4w + 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 + -0.4444444444w + w2 = 0

Move the constant term to the right:

Add '0.1111111111' to each side of the equation.
-0.1111111111 + -0.4444444444w + 0.1111111111 + w2 = 0 + 0.1111111111

Reorder the terms:
-0.1111111111 + 0.1111111111 + -0.4444444444w + w2 = 0 + 0.1111111111

Combine like terms: -0.1111111111 + 0.1111111111 = 0.0000000000
0.0000000000 + -0.4444444444w + w2 = 0 + 0.1111111111
-0.4444444444w + w2 = 0 + 0.1111111111

Combine like terms: 0 + 0.1111111111 = 0.1111111111
-0.4444444444w + w2 = 0.1111111111

The w term is -0.4444444444w.  Take half its coefficient (-0.2222222222).
Square it (0.04938271604) and add it to both sides.

Add '0.04938271604' to each side of the equation.
-0.4444444444w + 0.04938271604 + w2 = 0.1111111111 + 0.04938271604

Reorder the terms:
0.04938271604 + -0.4444444444w + w2 = 0.1111111111 + 0.04938271604

Combine like terms: 0.1111111111 + 0.04938271604 = 0.16049382714
0.04938271604 + -0.4444444444w + w2 = 0.16049382714

Factor a perfect square on the left side:
(w + -0.2222222222)(w + -0.2222222222) = 0.16049382714

Calculate the square root of the right side: 0.400616808

Break this problem into two subproblems by setting 
(w + -0.2222222222) equal to 0.400616808 and -0.400616808.

Subproblem 1
w + -0.2222222222 = 0.400616808

Simplifying
w + -0.2222222222 = 0.400616808

Reorder the terms:
-0.2222222222 + w = 0.400616808

Solving
-0.2222222222 + w = 0.400616808

Solving for variable 'w'.

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

Add '0.2222222222' to each side of the equation.
-0.2222222222 + 0.2222222222 + w = 0.400616808 + 0.2222222222

Combine like terms: -0.2222222222 + 0.2222222222 = 0.0000000000
0.0000000000 + w = 0.400616808 + 0.2222222222
w = 0.400616808 + 0.2222222222

Combine like terms: 0.400616808 + 0.2222222222 = 0.6228390302
w = 0.6228390302

Simplifying
w = 0.6228390302

Subproblem 2
w + -0.2222222222 = -0.400616808

Simplifying
w + -0.2222222222 = -0.400616808

Reorder the terms:
-0.2222222222 + w = -0.400616808

Solving
-0.2222222222 + w = -0.400616808

Solving for variable 'w'.

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

Add '0.2222222222' to each side of the equation.
-0.2222222222 + 0.2222222222 + w = -0.400616808 + 0.2222222222

Combine like terms: -0.2222222222 + 0.2222222222 = 0.0000000000
0.0000000000 + w = -0.400616808 + 0.2222222222
w = -0.400616808 + 0.2222222222

Combine like terms: -0.400616808 + 0.2222222222 = -0.1783945858
w = -0.1783945858

Simplifying
w = -0.1783945858

Solution
The solution to the problem is based on the solutions
from the subproblems.
w = {0.6228390302, -0.1783945858}