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Simplifying
w2 + 10.1w + 1 = 0

Reorder the terms:
1 + 10.1w + w2 = 0

Solving
1 + 10.1w + w2 = 0

Solving for variable 'w'.

Begin completing the square.

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + 10.1w + -1 + w2 = 0 + -1

Reorder the terms:
1 + -1 + 10.1w + w2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + 10.1w + w2 = 0 + -1
10.1w + w2 = 0 + -1

Combine like terms: 0 + -1 = -1
10.1w + w2 = -1

The w term is 10.1w.  Take half its coefficient (5.05).
Square it (25.5025) and add it to both sides.

Add '25.5025' to each side of the equation.
10.1w + 25.5025 + w2 = -1 + 25.5025

Reorder the terms:
25.5025 + 10.1w + w2 = -1 + 25.5025

Combine like terms: -1 + 25.5025 = 24.5025
25.5025 + 10.1w + w2 = 24.5025

Factor a perfect square on the left side:
(w + 5.05)(w + 5.05) = 24.5025

Calculate the square root of the right side: 4.95

Break this problem into two subproblems by setting 
(w + 5.05) equal to 4.95 and -4.95.

Subproblem 1

w + 5.05 = 4.95 Simplifying w + 5.05 = 4.95 Reorder the terms: 5.05 + w = 4.95 Solving 5.05 + w = 4.95 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-5.05' to each side of the equation. 5.05 + -5.05 + w = 4.95 + -5.05 Combine like terms: 5.05 + -5.05 = 0.00 0.00 + w = 4.95 + -5.05 w = 4.95 + -5.05 Combine like terms: 4.95 + -5.05 = -0.1 w = -0.1 Simplifying w = -0.1

Subproblem 2

w + 5.05 = -4.95 Simplifying w + 5.05 = -4.95 Reorder the terms: 5.05 + w = -4.95 Solving 5.05 + w = -4.95 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-5.05' to each side of the equation. 5.05 + -5.05 + w = -4.95 + -5.05 Combine like terms: 5.05 + -5.05 = 0.00 0.00 + w = -4.95 + -5.05 w = -4.95 + -5.05 Combine like terms: -4.95 + -5.05 = -10 w = -10 Simplifying w = -10

Solution

The solution to the problem is based on the solutions from the subproblems. w = {-0.1, -10}
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