9w^2+3w-3=0

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Solution for 9w^2+3w-3=0 equation: 


Simplifying
9w2 + 3w + -3 = 0

Reorder the terms:
-3 + 3w + 9w2 = 0

Solving
-3 + 3w + 9w2 = 0

Solving for variable 'w'.

Factor out the Greatest Common Factor (GCF), '3'.
3(-1 + w + 3w2) = 0

Ignore the factor 3.

Subproblem 1
Set the factor '(-1 + w + 3w2)' equal to zero and attempt to solve:

Simplifying
-1 + w + 3w2 = 0

Solving
-1 + w + 3w2 = 0

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

Divide each side by '3'.
-0.3333333333 + 0.3333333333w + w2 = 0

Move the constant term to the right:

Add '0.3333333333' to each side of the equation.
-0.3333333333 + 0.3333333333w + 0.3333333333 + w2 = 0 + 0.3333333333

Reorder the terms:
-0.3333333333 + 0.3333333333 + 0.3333333333w + w2 = 0 + 0.3333333333

Combine like terms: -0.3333333333 + 0.3333333333 = 0.0000000000
0.0000000000 + 0.3333333333w + w2 = 0 + 0.3333333333
0.3333333333w + w2 = 0 + 0.3333333333

Combine like terms: 0 + 0.3333333333 = 0.3333333333
0.3333333333w + w2 = 0.3333333333

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.
0.3333333333w + 0.25 + w2 = 0.3333333333 + 0.25

Reorder the terms:
0.25 + 0.3333333333w + w2 = 0.3333333333 + 0.25

Combine like terms: 0.3333333333 + 0.25 = 0.5833333333
0.25 + 0.3333333333w + w2 = 0.5833333333

Factor a perfect square on the left side:
(w + 0.5)(w + 0.5) = 0.5833333333

Calculate the square root of the right side: 0.763762616

Break this problem into two subproblems by setting 
(w + 0.5) equal to 0.763762616 and -0.763762616.


Subproblem 1
w + 0.5 = 0.763762616

Simplifying
w + 0.5 = 0.763762616

Reorder the terms:
0.5 + w = 0.763762616

Solving
0.5 + w = 0.763762616

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 = 0.763762616 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + w = 0.763762616 + -0.5
w = 0.763762616 + -0.5

Combine like terms: 0.763762616 + -0.5 = 0.263762616
w = 0.263762616

Simplifying
w = 0.263762616


Subproblem 2
w + 0.5 = -0.763762616

Simplifying
w + 0.5 = -0.763762616

Reorder the terms:
0.5 + w = -0.763762616

Solving
0.5 + w = -0.763762616

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 = -0.763762616 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + w = -0.763762616 + -0.5
w = -0.763762616 + -0.5

Combine like terms: -0.763762616 + -0.5 = -1.263762616
w = -1.263762616

Simplifying
w = -1.263762616


Solution
The solution to the problem is based on the solutions
from the subproblems.
w = {0.263762616, -1.263762616}
Solution
w = {0.263762616, -1.263762616}

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