6x^2+10x+3=0

Solution for 6x^2+10x+3=0 equation:

Simplifying
6x2 + 10x + 3 = 0

Reorder the terms:
3 + 10x + 6x2 = 0

Solving
3 + 10x + 6x2 = 0

Solving for variable 'x'.

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

Divide each side by '6'.
0.5 + 1.666666667x + x2 = 0

Move the constant term to the right:

Add '-0.5' to each side of the equation.
0.5 + 1.666666667x + -0.5 + x2 = 0 + -0.5

Reorder the terms:
0.5 + -0.5 + 1.666666667x + x2 = 0 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + 1.666666667x + x2 = 0 + -0.5
1.666666667x + x2 = 0 + -0.5

Combine like terms: 0 + -0.5 = -0.5
1.666666667x + x2 = -0.5

The x term is 1.666666667x.  Take half its coefficient (0.8333333335).
Square it (0.6944444447) and add it to both sides.

Add '0.6944444447' to each side of the equation.
1.666666667x + 0.6944444447 + x2 = -0.5 + 0.6944444447

Reorder the terms:
0.6944444447 + 1.666666667x + x2 = -0.5 + 0.6944444447

Combine like terms: -0.5 + 0.6944444447 = 0.1944444447
0.6944444447 + 1.666666667x + x2 = 0.1944444447

Factor a perfect square on the left side:
(x + 0.8333333335)(x + 0.8333333335) = 0.1944444447

Calculate the square root of the right side: 0.440958552

Break this problem into two subproblems by setting 
(x + 0.8333333335) equal to 0.440958552 and -0.440958552.

Subproblem 1
x + 0.8333333335 = 0.440958552

Simplifying
x + 0.8333333335 = 0.440958552

Reorder the terms:
0.8333333335 + x = 0.440958552

Solving
0.8333333335 + x = 0.440958552

Solving for variable 'x'.

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

Add '-0.8333333335' to each side of the equation.
0.8333333335 + -0.8333333335 + x = 0.440958552 + -0.8333333335

Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000
0.0000000000 + x = 0.440958552 + -0.8333333335
x = 0.440958552 + -0.8333333335

Combine like terms: 0.440958552 + -0.8333333335 = -0.3923747815
x = -0.3923747815

Simplifying
x = -0.3923747815

Subproblem 2
x + 0.8333333335 = -0.440958552

Simplifying
x + 0.8333333335 = -0.440958552

Reorder the terms:
0.8333333335 + x = -0.440958552

Solving
0.8333333335 + x = -0.440958552

Solving for variable 'x'.

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

Add '-0.8333333335' to each side of the equation.
0.8333333335 + -0.8333333335 + x = -0.440958552 + -0.8333333335

Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000
0.0000000000 + x = -0.440958552 + -0.8333333335
x = -0.440958552 + -0.8333333335

Combine like terms: -0.440958552 + -0.8333333335 = -1.2742918855
x = -1.2742918855

Simplifying
x = -1.2742918855

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.3923747815, -1.2742918855}