3x^2+x-(.5)=0

Solution for 3x^2+x-(.5)=0 equation:

Simplifying
3x2 + x + -1(0.5) = 0

Multiply -1 * 0.5
3x2 + x + -0.5 = 0

Reorder the terms:
-0.5 + x + 3x2 = 0

Solving
-0.5 + x + 3x2 = 0

Solving for variable 'x'.

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

Divide each side by '3'.
-0.1666666667 + 0.3333333333x + x2 = 0

Move the constant term to the right:

Add '0.1666666667' to each side of the equation.
-0.1666666667 + 0.3333333333x + 0.1666666667 + x2 = 0 + 0.1666666667

Reorder the terms:
-0.1666666667 + 0.1666666667 + 0.3333333333x + x2 = 0 + 0.1666666667

Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000
0.0000000000 + 0.3333333333x + x2 = 0 + 0.1666666667
0.3333333333x + x2 = 0 + 0.1666666667

Combine like terms: 0 + 0.1666666667 = 0.1666666667
0.3333333333x + x2 = 0.1666666667

The x term is x.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
0.3333333333x + 0.25 + x2 = 0.1666666667 + 0.25

Reorder the terms:
0.25 + 0.3333333333x + x2 = 0.1666666667 + 0.25

Combine like terms: 0.1666666667 + 0.25 = 0.4166666667
0.25 + 0.3333333333x + x2 = 0.4166666667

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 0.4166666667

Calculate the square root of the right side: 0.645497224

Break this problem into two subproblems by setting 
(x + 0.5) equal to 0.645497224 and -0.645497224.

Subproblem 1
x + 0.5 = 0.645497224

Simplifying
x + 0.5 = 0.645497224

Reorder the terms:
0.5 + x = 0.645497224

Solving
0.5 + x = 0.645497224

Solving for variable 'x'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x = 0.645497224 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 0.645497224 + -0.5
x = 0.645497224 + -0.5

Combine like terms: 0.645497224 + -0.5 = 0.145497224
x = 0.145497224

Simplifying
x = 0.145497224

Subproblem 2
x + 0.5 = -0.645497224

Simplifying
x + 0.5 = -0.645497224

Reorder the terms:
0.5 + x = -0.645497224

Solving
0.5 + x = -0.645497224

Solving for variable 'x'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x = -0.645497224 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -0.645497224 + -0.5
x = -0.645497224 + -0.5

Combine like terms: -0.645497224 + -0.5 = -1.145497224
x = -1.145497224

Simplifying
x = -1.145497224

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.145497224, -1.145497224}