5x^2+13x+1=0

Solution for 5x^2+13x+1=0 equation:

Simplifying
5x2 + 13x + 1 = 0

Reorder the terms:
1 + 13x + 5x2 = 0

Solving
1 + 13x + 5x2 = 0

Solving for variable 'x'.

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

Divide each side by '5'.
0.2 + 2.6x + x2 = 0

Move the constant term to the right:

Add '-0.2' to each side of the equation.
0.2 + 2.6x + -0.2 + x2 = 0 + -0.2

Reorder the terms:
0.2 + -0.2 + 2.6x + x2 = 0 + -0.2

Combine like terms: 0.2 + -0.2 = 0.0
0.0 + 2.6x + x2 = 0 + -0.2
2.6x + x2 = 0 + -0.2

Combine like terms: 0 + -0.2 = -0.2
2.6x + x2 = -0.2

The x term is 2.6x.  Take half its coefficient (1.3).
Square it (1.69) and add it to both sides.

Add '1.69' to each side of the equation.
2.6x + 1.69 + x2 = -0.2 + 1.69

Reorder the terms:
1.69 + 2.6x + x2 = -0.2 + 1.69

Combine like terms: -0.2 + 1.69 = 1.49
1.69 + 2.6x + x2 = 1.49

Factor a perfect square on the left side:
(x + 1.3)(x + 1.3) = 1.49

Calculate the square root of the right side: 1.220655562

Break this problem into two subproblems by setting 
(x + 1.3) equal to 1.220655562 and -1.220655562.

Subproblem 1
x + 1.3 = 1.220655562

Simplifying
x + 1.3 = 1.220655562

Reorder the terms:
1.3 + x = 1.220655562

Solving
1.3 + x = 1.220655562

Solving for variable 'x'.

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

Add '-1.3' to each side of the equation.
1.3 + -1.3 + x = 1.220655562 + -1.3

Combine like terms: 1.3 + -1.3 = 0.0
0.0 + x = 1.220655562 + -1.3
x = 1.220655562 + -1.3

Combine like terms: 1.220655562 + -1.3 = -0.079344438
x = -0.079344438

Simplifying
x = -0.079344438

Subproblem 2
x + 1.3 = -1.220655562

Simplifying
x + 1.3 = -1.220655562

Reorder the terms:
1.3 + x = -1.220655562

Solving
1.3 + x = -1.220655562

Solving for variable 'x'.

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

Add '-1.3' to each side of the equation.
1.3 + -1.3 + x = -1.220655562 + -1.3

Combine like terms: 1.3 + -1.3 = 0.0
0.0 + x = -1.220655562 + -1.3
x = -1.220655562 + -1.3

Combine like terms: -1.220655562 + -1.3 = -2.520655562
x = -2.520655562

Simplifying
x = -2.520655562

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.079344438, -2.520655562}