6x^2-x-21=0

Solution for 6x^2-x-21=0 equation:

Simplifying
6x2 + -1x + -21 = 0

Reorder the terms:
-21 + -1x + 6x2 = 0

Solving
-21 + -1x + 6x2 = 0

Solving for variable 'x'.

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

Divide each side by '6'.
-3.5 + -0.1666666667x + x2 = 0

Move the constant term to the right:

Add '3.5' to each side of the equation.
-3.5 + -0.1666666667x + 3.5 + x2 = 0 + 3.5

Reorder the terms:
-3.5 + 3.5 + -0.1666666667x + x2 = 0 + 3.5

Combine like terms: -3.5 + 3.5 = 0.0
0.0 + -0.1666666667x + x2 = 0 + 3.5
-0.1666666667x + x2 = 0 + 3.5

Combine like terms: 0 + 3.5 = 3.5
-0.1666666667x + x2 = 3.5

The x term is -0.1666666667x.  Take half its coefficient (-0.08333333335).
Square it (0.006944444447) and add it to both sides.

Add '0.006944444447' to each side of the equation.
-0.1666666667x + 0.006944444447 + x2 = 3.5 + 0.006944444447

Reorder the terms:
0.006944444447 + -0.1666666667x + x2 = 3.5 + 0.006944444447

Combine like terms: 3.5 + 0.006944444447 = 3.506944444447
0.006944444447 + -0.1666666667x + x2 = 3.506944444447

Factor a perfect square on the left side:
(x + -0.08333333335)(x + -0.08333333335) = 3.506944444447

Calculate the square root of the right side: 1.872683755

Break this problem into two subproblems by setting 
(x + -0.08333333335) equal to 1.872683755 and -1.872683755.

Subproblem 1
x + -0.08333333335 = 1.872683755

Simplifying
x + -0.08333333335 = 1.872683755

Reorder the terms:
-0.08333333335 + x = 1.872683755

Solving
-0.08333333335 + x = 1.872683755

Solving for variable 'x'.

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

Add '0.08333333335' to each side of the equation.
-0.08333333335 + 0.08333333335 + x = 1.872683755 + 0.08333333335

Combine like terms: -0.08333333335 + 0.08333333335 = 0.00000000000
0.00000000000 + x = 1.872683755 + 0.08333333335
x = 1.872683755 + 0.08333333335

Combine like terms: 1.872683755 + 0.08333333335 = 1.95601708835
x = 1.95601708835

Simplifying
x = 1.95601708835

Subproblem 2
x + -0.08333333335 = -1.872683755

Simplifying
x + -0.08333333335 = -1.872683755

Reorder the terms:
-0.08333333335 + x = -1.872683755

Solving
-0.08333333335 + x = -1.872683755

Solving for variable 'x'.

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

Add '0.08333333335' to each side of the equation.
-0.08333333335 + 0.08333333335 + x = -1.872683755 + 0.08333333335

Combine like terms: -0.08333333335 + 0.08333333335 = 0.00000000000
0.00000000000 + x = -1.872683755 + 0.08333333335
x = -1.872683755 + 0.08333333335

Combine like terms: -1.872683755 + 0.08333333335 = -1.78935042165
x = -1.78935042165

Simplifying
x = -1.78935042165

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.95601708835, -1.78935042165}