5v^2+9v+3=0

Solution for 5v^2+9v+3=0 equation:

Simplifying
5v2 + 9v + 3 = 0

Reorder the terms:
3 + 9v + 5v2 = 0

Solving
3 + 9v + 5v2 = 0

Solving for variable 'v'.

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

Divide each side by '5'.
0.6 + 1.8v + v2 = 0

Move the constant term to the right:

Add '-0.6' to each side of the equation.
0.6 + 1.8v + -0.6 + v2 = 0 + -0.6

Reorder the terms:
0.6 + -0.6 + 1.8v + v2 = 0 + -0.6

Combine like terms: 0.6 + -0.6 = 0.0
0.0 + 1.8v + v2 = 0 + -0.6
1.8v + v2 = 0 + -0.6

Combine like terms: 0 + -0.6 = -0.6
1.8v + v2 = -0.6

The v term is 1.8v.  Take half its coefficient (0.9).
Square it (0.81) and add it to both sides.

Add '0.81' to each side of the equation.
1.8v + 0.81 + v2 = -0.6 + 0.81

Reorder the terms:
0.81 + 1.8v + v2 = -0.6 + 0.81

Combine like terms: -0.6 + 0.81 = 0.21
0.81 + 1.8v + v2 = 0.21

Factor a perfect square on the left side:
(v + 0.9)(v + 0.9) = 0.21

Calculate the square root of the right side: 0.458257569

Break this problem into two subproblems by setting 
(v + 0.9) equal to 0.458257569 and -0.458257569.

Subproblem 1
v + 0.9 = 0.458257569

Simplifying
v + 0.9 = 0.458257569

Reorder the terms:
0.9 + v = 0.458257569

Solving
0.9 + v = 0.458257569

Solving for variable 'v'.

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

Add '-0.9' to each side of the equation.
0.9 + -0.9 + v = 0.458257569 + -0.9

Combine like terms: 0.9 + -0.9 = 0.0
0.0 + v = 0.458257569 + -0.9
v = 0.458257569 + -0.9

Combine like terms: 0.458257569 + -0.9 = -0.441742431
v = -0.441742431

Simplifying
v = -0.441742431

Subproblem 2
v + 0.9 = -0.458257569

Simplifying
v + 0.9 = -0.458257569

Reorder the terms:
0.9 + v = -0.458257569

Solving
0.9 + v = -0.458257569

Solving for variable 'v'.

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

Add '-0.9' to each side of the equation.
0.9 + -0.9 + v = -0.458257569 + -0.9

Combine like terms: 0.9 + -0.9 = 0.0
0.0 + v = -0.458257569 + -0.9
v = -0.458257569 + -0.9

Combine like terms: -0.458257569 + -0.9 = -1.358257569
v = -1.358257569

Simplifying
v = -1.358257569

Solution
The solution to the problem is based on the solutions
from the subproblems.
v = {-0.441742431, -1.358257569}