2v^2+4v+1=0

Solution for 2v^2+4v+1=0 equation:

Simplifying
2v2 + 4v + 1 = 0

Reorder the terms:
1 + 4v + 2v2 = 0

Solving
1 + 4v + 2v2 = 0

Solving for variable 'v'.

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

Divide each side by '2'.
0.5 + 2v + v2 = 0

Move the constant term to the right:

Add '-0.5' to each side of the equation.
0.5 + 2v + -0.5 + v2 = 0 + -0.5

Reorder the terms:
0.5 + -0.5 + 2v + v2 = 0 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + 2v + v2 = 0 + -0.5
2v + v2 = 0 + -0.5

Combine like terms: 0 + -0.5 = -0.5
2v + v2 = -0.5

The v term is 2v.  Take half its coefficient (1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
2v + 1 + v2 = -0.5 + 1

Reorder the terms:
1 + 2v + v2 = -0.5 + 1

Combine like terms: -0.5 + 1 = 0.5
1 + 2v + v2 = 0.5

Factor a perfect square on the left side:
(v + 1)(v + 1) = 0.5

Calculate the square root of the right side: 0.707106781

Break this problem into two subproblems by setting 
(v + 1) equal to 0.707106781 and -0.707106781.

Subproblem 1
v + 1 = 0.707106781

Simplifying
v + 1 = 0.707106781

Reorder the terms:
1 + v = 0.707106781

Solving
1 + v = 0.707106781

Solving for variable 'v'.

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

Add '-1' to each side of the equation.
1 + -1 + v = 0.707106781 + -1

Combine like terms: 1 + -1 = 0
0 + v = 0.707106781 + -1
v = 0.707106781 + -1

Combine like terms: 0.707106781 + -1 = -0.292893219
v = -0.292893219

Simplifying
v = -0.292893219

Subproblem 2
v + 1 = -0.707106781

Simplifying
v + 1 = -0.707106781

Reorder the terms:
1 + v = -0.707106781

Solving
1 + v = -0.707106781

Solving for variable 'v'.

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

Add '-1' to each side of the equation.
1 + -1 + v = -0.707106781 + -1

Combine like terms: 1 + -1 = 0
0 + v = -0.707106781 + -1
v = -0.707106781 + -1

Combine like terms: -0.707106781 + -1 = -1.707106781
v = -1.707106781

Simplifying
v = -1.707106781

Solution
The solution to the problem is based on the solutions
from the subproblems.
v = {-0.292893219, -1.707106781}