0.5(x)(x+2)=24

Solution for 0.5(x)(x+2)=24 equation:

Simplifying
0.5(x)(x + 2) = 24

Reorder the terms:
0.5x(2 + x) = 24
(2 * 0.5x + x * 0.5x) = 24
(1x + 0.5x2) = 24

Solving
1x + 0.5x2 = 24

Solving for variable 'x'.

Reorder the terms:
-24 + 1x + 0.5x2 = 24 + -24

Combine like terms: 24 + -24 = 0
-24 + 1x + 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'.
-48 + 2x + x2 = 0

Move the constant term to the right:

Add '48' to each side of the equation.
-48 + 2x + 48 + x2 = 0 + 48

Reorder the terms:
-48 + 48 + 2x + x2 = 0 + 48

Combine like terms: -48 + 48 = 0
0 + 2x + x2 = 0 + 48
2x + x2 = 0 + 48

Combine like terms: 0 + 48 = 48
2x + x2 = 48

The x term is 2x.  Take half its coefficient (1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
2x + 1 + x2 = 48 + 1

Reorder the terms:
1 + 2x + x2 = 48 + 1

Combine like terms: 48 + 1 = 49
1 + 2x + x2 = 49

Factor a perfect square on the left side:
(x + 1)(x + 1) = 49

Calculate the square root of the right side: 7

Break this problem into two subproblems by setting 
(x + 1) equal to 7 and -7.

Subproblem 1
x + 1 = 7

Simplifying
x + 1 = 7

Reorder the terms:
1 + x = 7

Solving
1 + x = 7

Solving for variable 'x'.

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

Add '-1' to each side of the equation.
1 + -1 + x = 7 + -1

Combine like terms: 1 + -1 = 0
0 + x = 7 + -1
x = 7 + -1

Combine like terms: 7 + -1 = 6
x = 6

Simplifying
x = 6

Subproblem 2
x + 1 = -7

Simplifying
x + 1 = -7

Reorder the terms:
1 + x = -7

Solving
1 + x = -7

Solving for variable 'x'.

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

Add '-1' to each side of the equation.
1 + -1 + x = -7 + -1

Combine like terms: 1 + -1 = 0
0 + x = -7 + -1
x = -7 + -1

Combine like terms: -7 + -1 = -8
x = -8

Simplifying
x = -8

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {6, -8}