Simplifying
w^{2} + 10.1w + 1 = 0
Reorder the terms:
1 + 10.1w + w^{2} = 0
Solving
1 + 10.1w + w^{2} = 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 + w^{2} = 0 + -1
Reorder the terms:
1 + -1 + 10.1w + w^{2} = 0 + -1
Combine like terms: 1 + -1 = 0
0 + 10.1w + w^{2} = 0 + -1
10.1w + w^{2} = 0 + -1
Combine like terms: 0 + -1 = -1
10.1w + w^{2} = -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 + w^{2} = -1 + 25.5025
Reorder the terms:
25.5025 + 10.1w + w^{2} = -1 + 25.5025
Combine like terms: -1 + 25.5025 = 24.5025
25.5025 + 10.1w + w^{2} = 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}