2v^2+4v+3=5

Solution for 2v^2+4v+3=5 equation:

Simplifying
2v2 + 4v + 3 = 5

Reorder the terms:
3 + 4v + 2v2 = 5

Solving
3 + 4v + 2v2 = 5

Solving for variable 'v'.

Reorder the terms:
3 + -5 + 4v + 2v2 = 5 + -5

Combine like terms: 3 + -5 = -2
-2 + 4v + 2v2 = 5 + -5

Combine like terms: 5 + -5 = 0
-2 + 4v + 2v2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-1 + 2v + v2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-1 + 2v + v2)' equal to zero and attempt to solve:

Simplifying
-1 + 2v + v2 = 0

Solving
-1 + 2v + v2 = 0

Begin completing the square.

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + 2v + 1 + v2 = 0 + 1

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

Combine like terms: -1 + 1 = 0
0 + 2v + v2 = 0 + 1
2v + v2 = 0 + 1

Combine like terms: 0 + 1 = 1
2v + v2 = 1

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 = 1 + 1

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

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

Factor a perfect square on the left side:
(v + 1)(v + 1) = 2

Calculate the square root of the right side: 1.414213562

Break this problem into two subproblems by setting 
(v + 1) equal to 1.414213562 and -1.414213562.

Subproblem 1
v + 1 = 1.414213562

Simplifying
v + 1 = 1.414213562

Reorder the terms:
1 + v = 1.414213562

Solving
1 + v = 1.414213562

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 = 1.414213562 + -1

Combine like terms: 1 + -1 = 0
0 + v = 1.414213562 + -1
v = 1.414213562 + -1

Combine like terms: 1.414213562 + -1 = 0.414213562
v = 0.414213562

Simplifying
v = 0.414213562

Subproblem 2
v + 1 = -1.414213562

Simplifying
v + 1 = -1.414213562

Reorder the terms:
1 + v = -1.414213562

Solving
1 + v = -1.414213562

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 = -1.414213562 + -1

Combine like terms: 1 + -1 = 0
0 + v = -1.414213562 + -1
v = -1.414213562 + -1

Combine like terms: -1.414213562 + -1 = -2.414213562
v = -2.414213562

Simplifying
v = -2.414213562

Solution
The solution to the problem is based on the solutions
from the subproblems.
v = {0.414213562, -2.414213562}
Solution
v = {0.414213562, -2.414213562}