w^2=3w-1

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


Simplifying
w2 = 3w + -1

Reorder the terms:
w2 = -1 + 3w

Solving
w2 = -1 + 3w

Solving for variable 'w'.

Reorder the terms:
1 + -3w + w2 = -1 + 3w + 1 + -3w

Reorder the terms:
1 + -3w + w2 = -1 + 1 + 3w + -3w

Combine like terms: -1 + 1 = 0
1 + -3w + w2 = 0 + 3w + -3w
1 + -3w + w2 = 3w + -3w

Combine like terms: 3w + -3w = 0
1 + -3w + w2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + -3w + -1 + w2 = 0 + -1

Reorder the terms:
1 + -1 + -3w + w2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + -3w + w2 = 0 + -1
-3w + w2 = 0 + -1

Combine like terms: 0 + -1 = -1
-3w + w2 = -1

The w term is -3w.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3w + 2.25 + w2 = -1 + 2.25

Reorder the terms:
2.25 + -3w + w2 = -1 + 2.25

Combine like terms: -1 + 2.25 = 1.25
2.25 + -3w + w2 = 1.25

Factor a perfect square on the left side:
(w + -1.5)(w + -1.5) = 1.25

Calculate the square root of the right side: 1.118033989

Break this problem into two subproblems by setting 
(w + -1.5) equal to 1.118033989 and -1.118033989.

Subproblem 1

w + -1.5 = 1.118033989 Simplifying w + -1.5 = 1.118033989 Reorder the terms: -1.5 + w = 1.118033989 Solving -1.5 + w = 1.118033989 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + w = 1.118033989 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + w = 1.118033989 + 1.5 w = 1.118033989 + 1.5 Combine like terms: 1.118033989 + 1.5 = 2.618033989 w = 2.618033989 Simplifying w = 2.618033989

Subproblem 2

w + -1.5 = -1.118033989 Simplifying w + -1.5 = -1.118033989 Reorder the terms: -1.5 + w = -1.118033989 Solving -1.5 + w = -1.118033989 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + w = -1.118033989 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + w = -1.118033989 + 1.5 w = -1.118033989 + 1.5 Combine like terms: -1.118033989 + 1.5 = 0.381966011 w = 0.381966011 Simplifying w = 0.381966011

Solution

The solution to the problem is based on the solutions from the subproblems. w = {2.618033989, 0.381966011}

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