0.6x^2+4x+0.5=0

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Solution for 0.6x^2+4x+0.5=0 equation: 


Simplifying
0.6x2 + 4x + 0.5 = 0

Reorder the terms:
0.5 + 4x + 0.6x2 = 0

Solving
0.5 + 4x + 0.6x2 = 0

Solving for variable 'x'.

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

Divide each side by '0.6'.
0.8333333333 + 6.666666667x + x2 = 0

Move the constant term to the right:

Add '-0.8333333333' to each side of the equation.
0.8333333333 + 6.666666667x + -0.8333333333 + x2 = 0 + -0.8333333333

Reorder the terms:
0.8333333333 + -0.8333333333 + 6.666666667x + x2 = 0 + -0.8333333333

Combine like terms: 0.8333333333 + -0.8333333333 = 0.0000000000
0.0000000000 + 6.666666667x + x2 = 0 + -0.8333333333
6.666666667x + x2 = 0 + -0.8333333333

Combine like terms: 0 + -0.8333333333 = -0.8333333333
6.666666667x + x2 = -0.8333333333

The x term is 6.666666667x.  Take half its coefficient (3.333333334).
Square it (11.11111112) and add it to both sides.

Add '11.11111112' to each side of the equation.
6.666666667x + 11.11111112 + x2 = -0.8333333333 + 11.11111112

Reorder the terms:
11.11111112 + 6.666666667x + x2 = -0.8333333333 + 11.11111112

Combine like terms: -0.8333333333 + 11.11111112 = 10.2777777867
11.11111112 + 6.666666667x + x2 = 10.2777777867

Factor a perfect square on the left side:
(x + 3.333333334)(x + 3.333333334) = 10.2777777867

Calculate the square root of the right side: 3.205897345

Break this problem into two subproblems by setting 
(x + 3.333333334) equal to 3.205897345 and -3.205897345.


Subproblem 1
x + 3.333333334 = 3.205897345

Simplifying
x + 3.333333334 = 3.205897345

Reorder the terms:
3.333333334 + x = 3.205897345

Solving
3.333333334 + x = 3.205897345

Solving for variable 'x'.

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

Add '-3.333333334' to each side of the equation.
3.333333334 + -3.333333334 + x = 3.205897345 + -3.333333334

Combine like terms: 3.333333334 + -3.333333334 = 0.000000000
0.000000000 + x = 3.205897345 + -3.333333334
x = 3.205897345 + -3.333333334

Combine like terms: 3.205897345 + -3.333333334 = -0.127435989
x = -0.127435989

Simplifying
x = -0.127435989


Subproblem 2
x + 3.333333334 = -3.205897345

Simplifying
x + 3.333333334 = -3.205897345

Reorder the terms:
3.333333334 + x = -3.205897345

Solving
3.333333334 + x = -3.205897345

Solving for variable 'x'.

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

Add '-3.333333334' to each side of the equation.
3.333333334 + -3.333333334 + x = -3.205897345 + -3.333333334

Combine like terms: 3.333333334 + -3.333333334 = 0.000000000
0.000000000 + x = -3.205897345 + -3.333333334
x = -3.205897345 + -3.333333334

Combine like terms: -3.205897345 + -3.333333334 = -6.539230679
x = -6.539230679

Simplifying
x = -6.539230679


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.127435989, -6.539230679}

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