5w^2+w+1=0

Solution for 5w^2+w+1=0 equation:

Simplifying
5w2 + w + 1 = 0

Reorder the terms:
1 + w + 5w2 = 0

Solving
1 + w + 5w2 = 0

Solving for variable 'w'.

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

Divide each side by '5'.
0.2 + 0.2w + w2 = 0

Move the constant term to the right:

Add '-0.2' to each side of the equation.
0.2 + 0.2w + -0.2 + w2 = 0 + -0.2

Reorder the terms:
0.2 + -0.2 + 0.2w + w2 = 0 + -0.2

Combine like terms: 0.2 + -0.2 = 0.0
0.0 + 0.2w + w2 = 0 + -0.2
0.2w + w2 = 0 + -0.2

Combine like terms: 0 + -0.2 = -0.2
0.2w + w2 = -0.2

The w term is w.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
0.2w + 0.25 + w2 = -0.2 + 0.25

Reorder the terms:
0.25 + 0.2w + w2 = -0.2 + 0.25

Combine like terms: -0.2 + 0.25 = 0.05
0.25 + 0.2w + w2 = 0.05

Factor a perfect square on the left side:
(w + 0.5)(w + 0.5) = 0.05

Calculate the square root of the right side: 0.223606798

Break this problem into two subproblems by setting 
(w + 0.5) equal to 0.223606798 and -0.223606798.

Subproblem 1
w + 0.5 = 0.223606798

Simplifying
w + 0.5 = 0.223606798

Reorder the terms:
0.5 + w = 0.223606798

Solving
0.5 + w = 0.223606798

Solving for variable 'w'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + w = 0.223606798 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + w = 0.223606798 + -0.5
w = 0.223606798 + -0.5

Combine like terms: 0.223606798 + -0.5 = -0.276393202
w = -0.276393202

Simplifying
w = -0.276393202

Subproblem 2
w + 0.5 = -0.223606798

Simplifying
w + 0.5 = -0.223606798

Reorder the terms:
0.5 + w = -0.223606798

Solving
0.5 + w = -0.223606798

Solving for variable 'w'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + w = -0.223606798 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + w = -0.223606798 + -0.5
w = -0.223606798 + -0.5

Combine like terms: -0.223606798 + -0.5 = -0.723606798
w = -0.723606798

Simplifying
w = -0.723606798

Solution
The solution to the problem is based on the solutions
from the subproblems.
w = {-0.276393202, -0.723606798}