0.5x^2+6x-21=0

Solution for 0.5x^2+6x-21=0 equation:

Simplifying
0.5x2 + 6x + -21 = 0

Reorder the terms:
-21 + 6x + 0.5x2 = 0

Solving
-21 + 6x + 0.5x2 = 0

Solving for variable 'x'.

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

Divide each side by '0.5'.
-42 + 12x + x2 = 0

Move the constant term to the right:

Add '42' to each side of the equation.
-42 + 12x + 42 + x2 = 0 + 42

Reorder the terms:
-42 + 42 + 12x + x2 = 0 + 42

Combine like terms: -42 + 42 = 0
0 + 12x + x2 = 0 + 42
12x + x2 = 0 + 42

Combine like terms: 0 + 42 = 42
12x + x2 = 42

The x term is 12x.  Take half its coefficient (6).
Square it (36) and add it to both sides.

Add '36' to each side of the equation.
12x + 36 + x2 = 42 + 36

Reorder the terms:
36 + 12x + x2 = 42 + 36

Combine like terms: 42 + 36 = 78
36 + 12x + x2 = 78

Factor a perfect square on the left side:
(x + 6)(x + 6) = 78

Calculate the square root of the right side: 8.831760866

Break this problem into two subproblems by setting 
(x + 6) equal to 8.831760866 and -8.831760866.

Subproblem 1
x + 6 = 8.831760866

Simplifying
x + 6 = 8.831760866

Reorder the terms:
6 + x = 8.831760866

Solving
6 + x = 8.831760866

Solving for variable 'x'.

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

Add '-6' to each side of the equation.
6 + -6 + x = 8.831760866 + -6

Combine like terms: 6 + -6 = 0
0 + x = 8.831760866 + -6
x = 8.831760866 + -6

Combine like terms: 8.831760866 + -6 = 2.831760866
x = 2.831760866

Simplifying
x = 2.831760866

Subproblem 2
x + 6 = -8.831760866

Simplifying
x + 6 = -8.831760866

Reorder the terms:
6 + x = -8.831760866

Solving
6 + x = -8.831760866

Solving for variable 'x'.

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

Add '-6' to each side of the equation.
6 + -6 + x = -8.831760866 + -6

Combine like terms: 6 + -6 = 0
0 + x = -8.831760866 + -6
x = -8.831760866 + -6

Combine like terms: -8.831760866 + -6 = -14.831760866
x = -14.831760866

Simplifying
x = -14.831760866

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.831760866, -14.831760866}