(0.5x)(x+1)=10

Solution for (0.5x)(x+1)=10 equation:

Simplifying
(0.5x)(x + 1) = 10

Remove parenthesis around (0.5x)
0.5x(x + 1) = 10

Reorder the terms:
0.5x(1 + x) = 10
(1 * 0.5x + x * 0.5x) = 10
(0.5x + 0.5x2) = 10

Solving
0.5x + 0.5x2 = 10

Solving for variable 'x'.

Reorder the terms:
-10 + 0.5x + 0.5x2 = 10 + -10

Combine like terms: 10 + -10 = 0
-10 + 0.5x + 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'.
-20 + x + x2 = 0

Move the constant term to the right:

Add '20' to each side of the equation.
-20 + x + 20 + x2 = 0 + 20

Reorder the terms:
-20 + 20 + x + x2 = 0 + 20

Combine like terms: -20 + 20 = 0
0 + x + x2 = 0 + 20
x + x2 = 0 + 20

Combine like terms: 0 + 20 = 20
x + x2 = 20

The x term is x.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
 + 0.25 + x2 = 20 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + x2 = 20 + 0.25

Combine like terms: 20 + 0.25 = 20.25
1.25 + x2 = 20.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 20.25

Calculate the square root of the right side: 4.5

Break this problem into two subproblems by setting 
(x + 0.5) equal to 4.5 and -4.5.

Subproblem 1
x + 0.5 = 4.5

Simplifying
x + 0.5 = 4.5

Reorder the terms:
0.5 + x = 4.5

Solving
0.5 + x = 4.5

Solving for variable 'x'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x = 4.5 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 4.5 + -0.5
x = 4.5 + -0.5

Combine like terms: 4.5 + -0.5 = 4
x = 4

Simplifying
x = 4

Subproblem 2
x + 0.5 = -4.5

Simplifying
x + 0.5 = -4.5

Reorder the terms:
0.5 + x = -4.5

Solving
0.5 + x = -4.5

Solving for variable 'x'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x = -4.5 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -4.5 + -0.5
x = -4.5 + -0.5

Combine like terms: -4.5 + -0.5 = -5
x = -5

Simplifying
x = -5

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {4, -5}