14x^2+19x+1=0

Solution for 14x^2+19x+1=0 equation:

Simplifying
14x2 + 19x + 1 = 0

Reorder the terms:
1 + 19x + 14x2 = 0

Solving
1 + 19x + 14x2 = 0

Solving for variable 'x'.

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

Divide each side by '14'.
0.07142857143 + 1.357142857x + x2 = 0

Move the constant term to the right:

Add '-0.07142857143' to each side of the equation.
0.07142857143 + 1.357142857x + -0.07142857143 + x2 = 0 + -0.07142857143

Reorder the terms:
0.07142857143 + -0.07142857143 + 1.357142857x + x2 = 0 + -0.07142857143

Combine like terms: 0.07142857143 + -0.07142857143 = 0.00000000000
0.00000000000 + 1.357142857x + x2 = 0 + -0.07142857143
1.357142857x + x2 = 0 + -0.07142857143

Combine like terms: 0 + -0.07142857143 = -0.07142857143
1.357142857x + x2 = -0.07142857143

The x term is 1.357142857x.  Take half its coefficient (0.6785714285).
Square it (0.4604591836) and add it to both sides.

Add '0.4604591836' to each side of the equation.
1.357142857x + 0.4604591836 + x2 = -0.07142857143 + 0.4604591836

Reorder the terms:
0.4604591836 + 1.357142857x + x2 = -0.07142857143 + 0.4604591836

Combine like terms: -0.07142857143 + 0.4604591836 = 0.38903061217
0.4604591836 + 1.357142857x + x2 = 0.38903061217

Factor a perfect square on the left side:
(x + 0.6785714285)(x + 0.6785714285) = 0.38903061217

Calculate the square root of the right side: 0.623723186

Break this problem into two subproblems by setting 
(x + 0.6785714285) equal to 0.623723186 and -0.623723186.

Subproblem 1
x + 0.6785714285 = 0.623723186

Simplifying
x + 0.6785714285 = 0.623723186

Reorder the terms:
0.6785714285 + x = 0.623723186

Solving
0.6785714285 + x = 0.623723186

Solving for variable 'x'.

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

Add '-0.6785714285' to each side of the equation.
0.6785714285 + -0.6785714285 + x = 0.623723186 + -0.6785714285

Combine like terms: 0.6785714285 + -0.6785714285 = 0.0000000000
0.0000000000 + x = 0.623723186 + -0.6785714285
x = 0.623723186 + -0.6785714285

Combine like terms: 0.623723186 + -0.6785714285 = -0.0548482425
x = -0.0548482425

Simplifying
x = -0.0548482425

Subproblem 2
x + 0.6785714285 = -0.623723186

Simplifying
x + 0.6785714285 = -0.623723186

Reorder the terms:
0.6785714285 + x = -0.623723186

Solving
0.6785714285 + x = -0.623723186

Solving for variable 'x'.

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

Add '-0.6785714285' to each side of the equation.
0.6785714285 + -0.6785714285 + x = -0.623723186 + -0.6785714285

Combine like terms: 0.6785714285 + -0.6785714285 = 0.0000000000
0.0000000000 + x = -0.623723186 + -0.6785714285
x = -0.623723186 + -0.6785714285

Combine like terms: -0.623723186 + -0.6785714285 = -1.3022946145
x = -1.3022946145

Simplifying
x = -1.3022946145

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.0548482425, -1.3022946145}