9w^2-12w-1=0

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Solution for 9w^2-12w-1=0 equation: 


Simplifying
9w2 + -12w + -1 = 0

Reorder the terms:
-1 + -12w + 9w2 = 0

Solving
-1 + -12w + 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 + -1.333333333w + w2 = 0

Move the constant term to the right:

Add '0.1111111111' to each side of the equation.
-0.1111111111 + -1.333333333w + 0.1111111111 + w2 = 0 + 0.1111111111

Reorder the terms:
-0.1111111111 + 0.1111111111 + -1.333333333w + w2 = 0 + 0.1111111111

Combine like terms: -0.1111111111 + 0.1111111111 = 0.0000000000
0.0000000000 + -1.333333333w + w2 = 0 + 0.1111111111
-1.333333333w + w2 = 0 + 0.1111111111

Combine like terms: 0 + 0.1111111111 = 0.1111111111
-1.333333333w + w2 = 0.1111111111

The w term is -1.333333333w.  Take half its coefficient (-0.6666666665).
Square it (0.4444444442) and add it to both sides.

Add '0.4444444442' to each side of the equation.
-1.333333333w + 0.4444444442 + w2 = 0.1111111111 + 0.4444444442

Reorder the terms:
0.4444444442 + -1.333333333w + w2 = 0.1111111111 + 0.4444444442

Combine like terms: 0.1111111111 + 0.4444444442 = 0.5555555553
0.4444444442 + -1.333333333w + w2 = 0.5555555553

Factor a perfect square on the left side:
(w + -0.6666666665)(w + -0.6666666665) = 0.5555555553

Calculate the square root of the right side: 0.745355992

Break this problem into two subproblems by setting 
(w + -0.6666666665) equal to 0.745355992 and -0.745355992.


Subproblem 1
w + -0.6666666665 = 0.745355992

Simplifying
w + -0.6666666665 = 0.745355992

Reorder the terms:
-0.6666666665 + w = 0.745355992

Solving
-0.6666666665 + w = 0.745355992

Solving for variable 'w'.

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

Add '0.6666666665' to each side of the equation.
-0.6666666665 + 0.6666666665 + w = 0.745355992 + 0.6666666665

Combine like terms: -0.6666666665 + 0.6666666665 = 0.0000000000
0.0000000000 + w = 0.745355992 + 0.6666666665
w = 0.745355992 + 0.6666666665

Combine like terms: 0.745355992 + 0.6666666665 = 1.4120226585
w = 1.4120226585

Simplifying
w = 1.4120226585


Subproblem 2
w + -0.6666666665 = -0.745355992

Simplifying
w + -0.6666666665 = -0.745355992

Reorder the terms:
-0.6666666665 + w = -0.745355992

Solving
-0.6666666665 + w = -0.745355992

Solving for variable 'w'.

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

Add '0.6666666665' to each side of the equation.
-0.6666666665 + 0.6666666665 + w = -0.745355992 + 0.6666666665

Combine like terms: -0.6666666665 + 0.6666666665 = 0.0000000000
0.0000000000 + w = -0.745355992 + 0.6666666665
w = -0.745355992 + 0.6666666665

Combine like terms: -0.745355992 + 0.6666666665 = -0.0786893255
w = -0.0786893255

Simplifying
w = -0.0786893255


Solution
The solution to the problem is based on the solutions
from the subproblems.
w = {1.4120226585, -0.0786893255}

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