.5*x(3+x)=104

Solution for .5*x(3+x)=104 equation:

Simplifying
0.5x(3 + x) = 104
(3 * 0.5x + x * 0.5x) = 104
(1.5x + 0.5x2) = 104

Solving
1.5x + 0.5x2 = 104

Solving for variable 'x'.

Reorder the terms:
-104 + 1.5x + 0.5x2 = 104 + -104

Combine like terms: 104 + -104 = 0
-104 + 1.5x + 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'.
-208 + 3x + x2 = 0

Move the constant term to the right:

Add '208' to each side of the equation.
-208 + 3x + 208 + x2 = 0 + 208

Reorder the terms:
-208 + 208 + 3x + x2 = 0 + 208

Combine like terms: -208 + 208 = 0
0 + 3x + x2 = 0 + 208
3x + x2 = 0 + 208

Combine like terms: 0 + 208 = 208
3x + x2 = 208

The x term is 3x.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3x + 2.25 + x2 = 208 + 2.25

Reorder the terms:
2.25 + 3x + x2 = 208 + 2.25

Combine like terms: 208 + 2.25 = 210.25
2.25 + 3x + x2 = 210.25

Factor a perfect square on the left side:
(x + 1.5)(x + 1.5) = 210.25

Calculate the square root of the right side: 14.5

Break this problem into two subproblems by setting 
(x + 1.5) equal to 14.5 and -14.5.

Subproblem 1
x + 1.5 = 14.5

Simplifying
x + 1.5 = 14.5

Reorder the terms:
1.5 + x = 14.5

Solving
1.5 + x = 14.5

Solving for variable 'x'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + x = 14.5 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + x = 14.5 + -1.5
x = 14.5 + -1.5

Combine like terms: 14.5 + -1.5 = 13
x = 13

Simplifying
x = 13

Subproblem 2
x + 1.5 = -14.5

Simplifying
x + 1.5 = -14.5

Reorder the terms:
1.5 + x = -14.5

Solving
1.5 + x = -14.5

Solving for variable 'x'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + x = -14.5 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + x = -14.5 + -1.5
x = -14.5 + -1.5

Combine like terms: -14.5 + -1.5 = -16
x = -16

Simplifying
x = -16

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {13, -16}