4x^2+3x+1=.5

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Solution for 4x^2+3x+1=.5 equation: 


Simplifying
4x2 + 3x + 1 = 0.5

Reorder the terms:
1 + 3x + 4x2 = 0.5

Solving
1 + 3x + 4x2 = 0.5

Solving for variable 'x'.

Reorder the terms:
1 + -0.5 + 3x + 4x2 = 0.5 + -0.5

Combine like terms: 1 + -0.5 = 0.5
0.5 + 3x + 4x2 = 0.5 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.5 + 3x + 4x2 = 0.0

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

Divide each side by '4'.
0.125 + 0.75x + x2 = 0.0

Move the constant term to the right:

Add '-0.125' to each side of the equation.
0.125 + 0.75x + -0.125 + x2 = 0.0 + -0.125

Reorder the terms:
0.125 + -0.125 + 0.75x + x2 = 0.0 + -0.125

Combine like terms: 0.125 + -0.125 = 0.000
0.000 + 0.75x + x2 = 0.0 + -0.125
0.75x + x2 = 0.0 + -0.125

Combine like terms: 0.0 + -0.125 = -0.125
0.75x + x2 = -0.125

The x term is 0.75x.  Take half its coefficient (0.375).
Square it (0.140625) and add it to both sides.

Add '0.140625' to each side of the equation.
0.75x + 0.140625 + x2 = -0.125 + 0.140625

Reorder the terms:
0.140625 + 0.75x + x2 = -0.125 + 0.140625

Combine like terms: -0.125 + 0.140625 = 0.015625
0.140625 + 0.75x + x2 = 0.015625

Factor a perfect square on the left side:
(x + 0.375)(x + 0.375) = 0.015625

Calculate the square root of the right side: 0.125

Break this problem into two subproblems by setting 
(x + 0.375) equal to 0.125 and -0.125.


Subproblem 1
x + 0.375 = 0.125

Simplifying
x + 0.375 = 0.125

Reorder the terms:
0.375 + x = 0.125

Solving
0.375 + x = 0.125

Solving for variable 'x'.

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

Add '-0.375' to each side of the equation.
0.375 + -0.375 + x = 0.125 + -0.375

Combine like terms: 0.375 + -0.375 = 0.000
0.000 + x = 0.125 + -0.375
x = 0.125 + -0.375

Combine like terms: 0.125 + -0.375 = -0.25
x = -0.25

Simplifying
x = -0.25


Subproblem 2
x + 0.375 = -0.125

Simplifying
x + 0.375 = -0.125

Reorder the terms:
0.375 + x = -0.125

Solving
0.375 + x = -0.125

Solving for variable 'x'.

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

Add '-0.375' to each side of the equation.
0.375 + -0.375 + x = -0.125 + -0.375

Combine like terms: 0.375 + -0.375 = 0.000
0.000 + x = -0.125 + -0.375
x = -0.125 + -0.375

Combine like terms: -0.125 + -0.375 = -0.5
x = -0.5

Simplifying
x = -0.5


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.25, -0.5}

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