(7x^2)+x+1=0

Solution for (7x^2)+x+1=0 equation:

Simplifying
(7x2) + x + 1 = 0

Reorder the terms:
1 + x + (7x2) = 0

Solving
1 + x + (7x2) = 0

Solving for variable 'x'.

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

Divide each side by '7'.
0.1428571429 + 0.1428571429x + x2 = 0

Move the constant term to the right:

Add '-0.1428571429' to each side of the equation.
0.1428571429 + 0.1428571429x + -0.1428571429 + x2 = 0 + -0.1428571429

Reorder the terms:
0.1428571429 + -0.1428571429 + 0.1428571429x + x2 = 0 + -0.1428571429

Combine like terms: 0.1428571429 + -0.1428571429 = 0.0000000000
0.0000000000 + 0.1428571429x + x2 = 0 + -0.1428571429
0.1428571429x + x2 = 0 + -0.1428571429

Combine like terms: 0 + -0.1428571429 = -0.1428571429
0.1428571429x + x2 = -0.1428571429

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.
0.1428571429x + 0.25 + x2 = -0.1428571429 + 0.25

Reorder the terms:
0.25 + 0.1428571429x + x2 = -0.1428571429 + 0.25

Combine like terms: -0.1428571429 + 0.25 = 0.1071428571
0.25 + 0.1428571429x + x2 = 0.1071428571

Factor a perfect square on the left side:
((x) + 0.5)((x) + 0.5) = 0.1071428571

Calculate the square root of the right side: 0.327326835

Break this problem into two subproblems by setting 
((x) + 0.5) equal to 0.327326835 and -0.327326835.

Subproblem 1
(x) + 0.5 = 0.327326835

Simplifying
(x) + 0.5 = 0.327326835
x + 0.5 = 0.327326835

Reorder the terms:
0.5 + x = 0.327326835

Solving
0.5 + x = 0.327326835

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 = 0.327326835 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 0.327326835 + -0.5
x = 0.327326835 + -0.5

Combine like terms: 0.327326835 + -0.5 = -0.172673165
x = -0.172673165

Simplifying
x = -0.172673165

Subproblem 2
(x) + 0.5 = -0.327326835

Simplifying
(x) + 0.5 = -0.327326835
x + 0.5 = -0.327326835

Reorder the terms:
0.5 + x = -0.327326835

Solving
0.5 + x = -0.327326835

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 = -0.327326835 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -0.327326835 + -0.5
x = -0.327326835 + -0.5

Combine like terms: -0.327326835 + -0.5 = -0.827326835
x = -0.827326835

Simplifying
x = -0.827326835

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.172673165, -0.827326835}