13x^2-7x-5=0

Solution for 13x^2-7x-5=0 equation:

Simplifying
13x2 + -7x + -5 = 0

Reorder the terms:
-5 + -7x + 13x2 = 0

Solving
-5 + -7x + 13x2 = 0

Solving for variable 'x'.

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

Divide each side by '13'.
-0.3846153846 + -0.5384615385x + x2 = 0

Move the constant term to the right:

Add '0.3846153846' to each side of the equation.
-0.3846153846 + -0.5384615385x + 0.3846153846 + x2 = 0 + 0.3846153846

Reorder the terms:
-0.3846153846 + 0.3846153846 + -0.5384615385x + x2 = 0 + 0.3846153846

Combine like terms: -0.3846153846 + 0.3846153846 = 0.0000000000
0.0000000000 + -0.5384615385x + x2 = 0 + 0.3846153846
-0.5384615385x + x2 = 0 + 0.3846153846

Combine like terms: 0 + 0.3846153846 = 0.3846153846
-0.5384615385x + x2 = 0.3846153846

The x term is -0.5384615385x.  Take half its coefficient (-0.2692307693).
Square it (0.07248520714) and add it to both sides.

Add '0.07248520714' to each side of the equation.
-0.5384615385x + 0.07248520714 + x2 = 0.3846153846 + 0.07248520714

Reorder the terms:
0.07248520714 + -0.5384615385x + x2 = 0.3846153846 + 0.07248520714

Combine like terms: 0.3846153846 + 0.07248520714 = 0.45710059174
0.07248520714 + -0.5384615385x + x2 = 0.45710059174

Factor a perfect square on the left side:
(x + -0.2692307693)(x + -0.2692307693) = 0.45710059174

Calculate the square root of the right side: 0.676092147

Break this problem into two subproblems by setting 
(x + -0.2692307693) equal to 0.676092147 and -0.676092147.

Subproblem 1
x + -0.2692307693 = 0.676092147

Simplifying
x + -0.2692307693 = 0.676092147

Reorder the terms:
-0.2692307693 + x = 0.676092147

Solving
-0.2692307693 + x = 0.676092147

Solving for variable 'x'.

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

Add '0.2692307693' to each side of the equation.
-0.2692307693 + 0.2692307693 + x = 0.676092147 + 0.2692307693

Combine like terms: -0.2692307693 + 0.2692307693 = 0.0000000000
0.0000000000 + x = 0.676092147 + 0.2692307693
x = 0.676092147 + 0.2692307693

Combine like terms: 0.676092147 + 0.2692307693 = 0.9453229163
x = 0.9453229163

Simplifying
x = 0.9453229163

Subproblem 2
x + -0.2692307693 = -0.676092147

Simplifying
x + -0.2692307693 = -0.676092147

Reorder the terms:
-0.2692307693 + x = -0.676092147

Solving
-0.2692307693 + x = -0.676092147

Solving for variable 'x'.

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

Add '0.2692307693' to each side of the equation.
-0.2692307693 + 0.2692307693 + x = -0.676092147 + 0.2692307693

Combine like terms: -0.2692307693 + 0.2692307693 = 0.0000000000
0.0000000000 + x = -0.676092147 + 0.2692307693
x = -0.676092147 + 0.2692307693

Combine like terms: -0.676092147 + 0.2692307693 = -0.4068613777
x = -0.4068613777

Simplifying
x = -0.4068613777

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.9453229163, -0.4068613777}