.5(x+3)(x+6)=119

Solution for .5(x+3)(x+6)=119 equation:

Simplifying
0.5(x + 3)(x + 6) = 119

Reorder the terms:
0.5(3 + x)(x + 6) = 119

Reorder the terms:
0.5(3 + x)(6 + x) = 119

Multiply (3 + x) * (6 + x)
0.5(3(6 + x) + x(6 + x)) = 119
0.5((6 * 3 + x * 3) + x(6 + x)) = 119
0.5((18 + 3x) + x(6 + x)) = 119
0.5(18 + 3x + (6 * x + x * x)) = 119
0.5(18 + 3x + (6x + x2)) = 119

Combine like terms: 3x + 6x = 9x
0.5(18 + 9x + x2) = 119
(18 * 0.5 + 9x * 0.5 + x2 * 0.5) = 119
(9 + 4.5x + 0.5x2) = 119

Solving
9 + 4.5x + 0.5x2 = 119

Solving for variable 'x'.

Reorder the terms:
9 + -119 + 4.5x + 0.5x2 = 119 + -119

Combine like terms: 9 + -119 = -110
-110 + 4.5x + 0.5x2 = 119 + -119

Combine like terms: 119 + -119 = 0
-110 + 4.5x + 0.5x2 = 0

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

Divide each side by '0.5'.
-220 + 9x + x2 = 0

Move the constant term to the right:

Add '220' to each side of the equation.
-220 + 9x + 220 + x2 = 0 + 220

Reorder the terms:
-220 + 220 + 9x + x2 = 0 + 220

Combine like terms: -220 + 220 = 0
0 + 9x + x2 = 0 + 220
9x + x2 = 0 + 220

Combine like terms: 0 + 220 = 220
9x + x2 = 220

The x term is 9x.  Take half its coefficient (4.5).
Square it (20.25) and add it to both sides.

Add '20.25' to each side of the equation.
9x + 20.25 + x2 = 220 + 20.25

Reorder the terms:
20.25 + 9x + x2 = 220 + 20.25

Combine like terms: 220 + 20.25 = 240.25
20.25 + 9x + x2 = 240.25

Factor a perfect square on the left side:
(x + 4.5)(x + 4.5) = 240.25

Calculate the square root of the right side: 15.5

Break this problem into two subproblems by setting 
(x + 4.5) equal to 15.5 and -15.5.

Subproblem 1
x + 4.5 = 15.5

Simplifying
x + 4.5 = 15.5

Reorder the terms:
4.5 + x = 15.5

Solving
4.5 + x = 15.5

Solving for variable 'x'.

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

Add '-4.5' to each side of the equation.
4.5 + -4.5 + x = 15.5 + -4.5

Combine like terms: 4.5 + -4.5 = 0.0
0.0 + x = 15.5 + -4.5
x = 15.5 + -4.5

Combine like terms: 15.5 + -4.5 = 11
x = 11

Simplifying
x = 11

Subproblem 2
x + 4.5 = -15.5

Simplifying
x + 4.5 = -15.5

Reorder the terms:
4.5 + x = -15.5

Solving
4.5 + x = -15.5

Solving for variable 'x'.

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

Add '-4.5' to each side of the equation.
4.5 + -4.5 + x = -15.5 + -4.5

Combine like terms: 4.5 + -4.5 = 0.0
0.0 + x = -15.5 + -4.5
x = -15.5 + -4.5

Combine like terms: -15.5 + -4.5 = -20
x = -20

Simplifying
x = -20

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {11, -20}