45w^2-6w-5=0

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Solution for 45w^2-6w-5=0 equation: 


Simplifying
45w2 + -6w + -5 = 0

Reorder the terms:
-5 + -6w + 45w2 = 0

Solving
-5 + -6w + 45w2 = 0

Solving for variable 'w'.

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

Divide each side by '45'.
-0.1111111111 + -0.1333333333w + w2 = 0

Move the constant term to the right:

Add '0.1111111111' to each side of the equation.
-0.1111111111 + -0.1333333333w + 0.1111111111 + w2 = 0 + 0.1111111111

Reorder the terms:
-0.1111111111 + 0.1111111111 + -0.1333333333w + w2 = 0 + 0.1111111111

Combine like terms: -0.1111111111 + 0.1111111111 = 0.0000000000
0.0000000000 + -0.1333333333w + w2 = 0 + 0.1111111111
-0.1333333333w + w2 = 0 + 0.1111111111

Combine like terms: 0 + 0.1111111111 = 0.1111111111
-0.1333333333w + w2 = 0.1111111111

The w term is -0.1333333333w.  Take half its coefficient (-0.06666666665).
Square it (0.004444444442) and add it to both sides.

Add '0.004444444442' to each side of the equation.
-0.1333333333w + 0.004444444442 + w2 = 0.1111111111 + 0.004444444442

Reorder the terms:
0.004444444442 + -0.1333333333w + w2 = 0.1111111111 + 0.004444444442

Combine like terms: 0.1111111111 + 0.004444444442 = 0.115555555542
0.004444444442 + -0.1333333333w + w2 = 0.115555555542

Factor a perfect square on the left side:
(w + -0.06666666665)(w + -0.06666666665) = 0.115555555542

Calculate the square root of the right side: 0.339934634

Break this problem into two subproblems by setting 
(w + -0.06666666665) equal to 0.339934634 and -0.339934634.


Subproblem 1
w + -0.06666666665 = 0.339934634

Simplifying
w + -0.06666666665 = 0.339934634

Reorder the terms:
-0.06666666665 + w = 0.339934634

Solving
-0.06666666665 + w = 0.339934634

Solving for variable 'w'.

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

Add '0.06666666665' to each side of the equation.
-0.06666666665 + 0.06666666665 + w = 0.339934634 + 0.06666666665

Combine like terms: -0.06666666665 + 0.06666666665 = 0.00000000000
0.00000000000 + w = 0.339934634 + 0.06666666665
w = 0.339934634 + 0.06666666665

Combine like terms: 0.339934634 + 0.06666666665 = 0.40660130065
w = 0.40660130065

Simplifying
w = 0.40660130065


Subproblem 2
w + -0.06666666665 = -0.339934634

Simplifying
w + -0.06666666665 = -0.339934634

Reorder the terms:
-0.06666666665 + w = -0.339934634

Solving
-0.06666666665 + w = -0.339934634

Solving for variable 'w'.

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

Add '0.06666666665' to each side of the equation.
-0.06666666665 + 0.06666666665 + w = -0.339934634 + 0.06666666665

Combine like terms: -0.06666666665 + 0.06666666665 = 0.00000000000
0.00000000000 + w = -0.339934634 + 0.06666666665
w = -0.339934634 + 0.06666666665

Combine like terms: -0.339934634 + 0.06666666665 = -0.27326796735
w = -0.27326796735

Simplifying
w = -0.27326796735


Solution
The solution to the problem is based on the solutions
from the subproblems.
w = {0.40660130065, -0.27326796735}

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