4x^2+5x+0.5=0

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Solution for 4x^2+5x+0.5=0 equation: 


Simplifying
4x2 + 5x + 0.5 = 0

Reorder the terms:
0.5 + 5x + 4x2 = 0

Solving
0.5 + 5x + 4x2 = 0

Solving for variable 'x'.

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

Divide each side by '4'.
0.125 + 1.25x + x2 = 0

Move the constant term to the right:

Add '-0.125' to each side of the equation.
0.125 + 1.25x + -0.125 + x2 = 0 + -0.125

Reorder the terms:
0.125 + -0.125 + 1.25x + x2 = 0 + -0.125

Combine like terms: 0.125 + -0.125 = 0.000
0.000 + 1.25x + x2 = 0 + -0.125
1.25x + x2 = 0 + -0.125

Combine like terms: 0 + -0.125 = -0.125
1.25x + x2 = -0.125

The x term is 1.25x.  Take half its coefficient (0.625).
Square it (0.390625) and add it to both sides.

Add '0.390625' to each side of the equation.
1.25x + 0.390625 + x2 = -0.125 + 0.390625

Reorder the terms:
0.390625 + 1.25x + x2 = -0.125 + 0.390625

Combine like terms: -0.125 + 0.390625 = 0.265625
0.390625 + 1.25x + x2 = 0.265625

Factor a perfect square on the left side:
(x + 0.625)(x + 0.625) = 0.265625

Calculate the square root of the right side: 0.515388203

Break this problem into two subproblems by setting 
(x + 0.625) equal to 0.515388203 and -0.515388203.


Subproblem 1
x + 0.625 = 0.515388203

Simplifying
x + 0.625 = 0.515388203

Reorder the terms:
0.625 + x = 0.515388203

Solving
0.625 + x = 0.515388203

Solving for variable 'x'.

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

Add '-0.625' to each side of the equation.
0.625 + -0.625 + x = 0.515388203 + -0.625

Combine like terms: 0.625 + -0.625 = 0.000
0.000 + x = 0.515388203 + -0.625
x = 0.515388203 + -0.625

Combine like terms: 0.515388203 + -0.625 = -0.109611797
x = -0.109611797

Simplifying
x = -0.109611797


Subproblem 2
x + 0.625 = -0.515388203

Simplifying
x + 0.625 = -0.515388203

Reorder the terms:
0.625 + x = -0.515388203

Solving
0.625 + x = -0.515388203

Solving for variable 'x'.

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

Add '-0.625' to each side of the equation.
0.625 + -0.625 + x = -0.515388203 + -0.625

Combine like terms: 0.625 + -0.625 = 0.000
0.000 + x = -0.515388203 + -0.625
x = -0.515388203 + -0.625

Combine like terms: -0.515388203 + -0.625 = -1.140388203
x = -1.140388203

Simplifying
x = -1.140388203


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.109611797, -1.140388203}

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