5x^2+11x+3=0

Solution for 5x^2+11x+3=0 equation:

Simplifying
5x2 + 11x + 3 = 0

Reorder the terms:
3 + 11x + 5x2 = 0

Solving
3 + 11x + 5x2 = 0

Solving for variable 'x'.

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

Divide each side by '5'.
0.6 + 2.2x + x2 = 0

Move the constant term to the right:

Add '-0.6' to each side of the equation.
0.6 + 2.2x + -0.6 + x2 = 0 + -0.6

Reorder the terms:
0.6 + -0.6 + 2.2x + x2 = 0 + -0.6

Combine like terms: 0.6 + -0.6 = 0.0
0.0 + 2.2x + x2 = 0 + -0.6
2.2x + x2 = 0 + -0.6

Combine like terms: 0 + -0.6 = -0.6
2.2x + x2 = -0.6

The x term is 2.2x.  Take half its coefficient (1.1).
Square it (1.21) and add it to both sides.

Add '1.21' to each side of the equation.
2.2x + 1.21 + x2 = -0.6 + 1.21

Reorder the terms:
1.21 + 2.2x + x2 = -0.6 + 1.21

Combine like terms: -0.6 + 1.21 = 0.61
1.21 + 2.2x + x2 = 0.61

Factor a perfect square on the left side:
(x + 1.1)(x + 1.1) = 0.61

Calculate the square root of the right side: 0.781024968

Break this problem into two subproblems by setting 
(x + 1.1) equal to 0.781024968 and -0.781024968.

Subproblem 1
x + 1.1 = 0.781024968

Simplifying
x + 1.1 = 0.781024968

Reorder the terms:
1.1 + x = 0.781024968

Solving
1.1 + x = 0.781024968

Solving for variable 'x'.

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

Add '-1.1' to each side of the equation.
1.1 + -1.1 + x = 0.781024968 + -1.1

Combine like terms: 1.1 + -1.1 = 0.0
0.0 + x = 0.781024968 + -1.1
x = 0.781024968 + -1.1

Combine like terms: 0.781024968 + -1.1 = -0.318975032
x = -0.318975032

Simplifying
x = -0.318975032

Subproblem 2
x + 1.1 = -0.781024968

Simplifying
x + 1.1 = -0.781024968

Reorder the terms:
1.1 + x = -0.781024968

Solving
1.1 + x = -0.781024968

Solving for variable 'x'.

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

Add '-1.1' to each side of the equation.
1.1 + -1.1 + x = -0.781024968 + -1.1

Combine like terms: 1.1 + -1.1 = 0.0
0.0 + x = -0.781024968 + -1.1
x = -0.781024968 + -1.1

Combine like terms: -0.781024968 + -1.1 = -1.881024968
x = -1.881024968

Simplifying
x = -1.881024968

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.318975032, -1.881024968}