w^2=3(3w-1)

Solution for w^2=3(3w-1) equation:

Simplifying
w2 = 3(3w + -1)

Reorder the terms:
w2 = 3(-1 + 3w)
w2 = (-1 * 3 + 3w * 3)
w2 = (-3 + 9w)

Solving
w2 = -3 + 9w

Solving for variable 'w'.

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

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

Combine like terms: -3 + 3 = 0
3 + -9w + w2 = 0 + 9w + -9w
3 + -9w + w2 = 9w + -9w

Combine like terms: 9w + -9w = 0
3 + -9w + w2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-3' to each side of the equation.
3 + -9w + -3 + w2 = 0 + -3

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

Combine like terms: 3 + -3 = 0
0 + -9w + w2 = 0 + -3
-9w + w2 = 0 + -3

Combine like terms: 0 + -3 = -3
-9w + w2 = -3

The w term is -9w.  Take half its coefficient (-4.5).
Square it (20.25) and add it to both sides.

Add '20.25' to each side of the equation.
-9w + 20.25 + w2 = -3 + 20.25

Reorder the terms:
20.25 + -9w + w2 = -3 + 20.25

Combine like terms: -3 + 20.25 = 17.25
20.25 + -9w + w2 = 17.25

Factor a perfect square on the left side:
(w + -4.5)(w + -4.5) = 17.25

Calculate the square root of the right side: 4.153311931

Break this problem into two subproblems by setting 
(w + -4.5) equal to 4.153311931 and -4.153311931.

Subproblem 1
w + -4.5 = 4.153311931

Simplifying
w + -4.5 = 4.153311931

Reorder the terms:
-4.5 + w = 4.153311931

Solving
-4.5 + w = 4.153311931

Solving for variable 'w'.

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

Add '4.5' to each side of the equation.
-4.5 + 4.5 + w = 4.153311931 + 4.5

Combine like terms: -4.5 + 4.5 = 0.0
0.0 + w = 4.153311931 + 4.5
w = 4.153311931 + 4.5

Combine like terms: 4.153311931 + 4.5 = 8.653311931
w = 8.653311931

Simplifying
w = 8.653311931

Subproblem 2
w + -4.5 = -4.153311931

Simplifying
w + -4.5 = -4.153311931

Reorder the terms:
-4.5 + w = -4.153311931

Solving
-4.5 + w = -4.153311931

Solving for variable 'w'.

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

Add '4.5' to each side of the equation.
-4.5 + 4.5 + w = -4.153311931 + 4.5

Combine like terms: -4.5 + 4.5 = 0.0
0.0 + w = -4.153311931 + 4.5
w = -4.153311931 + 4.5

Combine like terms: -4.153311931 + 4.5 = 0.346688069
w = 0.346688069

Simplifying
w = 0.346688069

Solution
The solution to the problem is based on the solutions
from the subproblems.
w = {8.653311931, 0.346688069}