21x^2-13x-2=0

Solution for 21x^2-13x-2=0 equation:

Simplifying
21x2 + -13x + -2 = 0

Reorder the terms:
-2 + -13x + 21x2 = 0

Solving
-2 + -13x + 21x2 = 0

Solving for variable 'x'.

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

Divide each side by '21'.
-0.09523809524 + -0.619047619x + x2 = 0

Move the constant term to the right:

Add '0.09523809524' to each side of the equation.
-0.09523809524 + -0.619047619x + 0.09523809524 + x2 = 0 + 0.09523809524

Reorder the terms:
-0.09523809524 + 0.09523809524 + -0.619047619x + x2 = 0 + 0.09523809524

Combine like terms: -0.09523809524 + 0.09523809524 = 0.00000000000
0.00000000000 + -0.619047619x + x2 = 0 + 0.09523809524
-0.619047619x + x2 = 0 + 0.09523809524

Combine like terms: 0 + 0.09523809524 = 0.09523809524
-0.619047619x + x2 = 0.09523809524

The x term is -0.619047619x.  Take half its coefficient (-0.3095238095).
Square it (0.09580498865) and add it to both sides.

Add '0.09580498865' to each side of the equation.
-0.619047619x + 0.09580498865 + x2 = 0.09523809524 + 0.09580498865

Reorder the terms:
0.09580498865 + -0.619047619x + x2 = 0.09523809524 + 0.09580498865

Combine like terms: 0.09523809524 + 0.09580498865 = 0.19104308389
0.09580498865 + -0.619047619x + x2 = 0.19104308389

Factor a perfect square on the left side:
(x + -0.3095238095)(x + -0.3095238095) = 0.19104308389

Calculate the square root of the right side: 0.437084756

Break this problem into two subproblems by setting 
(x + -0.3095238095) equal to 0.437084756 and -0.437084756.

Subproblem 1
x + -0.3095238095 = 0.437084756

Simplifying
x + -0.3095238095 = 0.437084756

Reorder the terms:
-0.3095238095 + x = 0.437084756

Solving
-0.3095238095 + x = 0.437084756

Solving for variable 'x'.

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

Add '0.3095238095' to each side of the equation.
-0.3095238095 + 0.3095238095 + x = 0.437084756 + 0.3095238095

Combine like terms: -0.3095238095 + 0.3095238095 = 0.0000000000
0.0000000000 + x = 0.437084756 + 0.3095238095
x = 0.437084756 + 0.3095238095

Combine like terms: 0.437084756 + 0.3095238095 = 0.7466085655
x = 0.7466085655

Simplifying
x = 0.7466085655

Subproblem 2
x + -0.3095238095 = -0.437084756

Simplifying
x + -0.3095238095 = -0.437084756

Reorder the terms:
-0.3095238095 + x = -0.437084756

Solving
-0.3095238095 + x = -0.437084756

Solving for variable 'x'.

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

Add '0.3095238095' to each side of the equation.
-0.3095238095 + 0.3095238095 + x = -0.437084756 + 0.3095238095

Combine like terms: -0.3095238095 + 0.3095238095 = 0.0000000000
0.0000000000 + x = -0.437084756 + 0.3095238095
x = -0.437084756 + 0.3095238095

Combine like terms: -0.437084756 + 0.3095238095 = -0.1275609465
x = -0.1275609465

Simplifying
x = -0.1275609465

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.7466085655, -0.1275609465}