.5a(a+1)=a+2

Solution for .5a(a+1)=a+2 equation:

Simplifying
0.5a(a + 1) = a + 2

Reorder the terms:
0.5a(1 + a) = a + 2
(1 * 0.5a + a * 0.5a) = a + 2
(0.5a + 0.5a2) = a + 2

Reorder the terms:
0.5a + 0.5a2 = 2 + a

Solving
0.5a + 0.5a2 = 2 + a

Solving for variable 'a'.

Reorder the terms:
-2 + 0.5a + -1a + 0.5a2 = 2 + a + -2 + -1a

Combine like terms: 0.5a + -1a = -0.5a
-2 + -0.5a + 0.5a2 = 2 + a + -2 + -1a

Reorder the terms:
-2 + -0.5a + 0.5a2 = 2 + -2 + a + -1a

Combine like terms: 2 + -2 = 0
-2 + -0.5a + 0.5a2 = 0 + a + -1a
-2 + -0.5a + 0.5a2 = a + -1a

Combine like terms: a + -1a = 0
-2 + -0.5a + 0.5a2 = 0

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

Divide each side by '0.5'.
-4 + -1a + a2 = 0

Move the constant term to the right:

Add '4' to each side of the equation.
-4 + -1a + 4 + a2 = 0 + 4

Reorder the terms:
-4 + 4 + -1a + a2 = 0 + 4

Combine like terms: -4 + 4 = 0
0 + -1a + a2 = 0 + 4
-1a + a2 = 0 + 4

Combine like terms: 0 + 4 = 4
-1a + a2 = 4

The a term is -1a.  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.
-1a + 0.25 + a2 = 4 + 0.25

Reorder the terms:
0.25 + -1a + a2 = 4 + 0.25

Combine like terms: 4 + 0.25 = 4.25
0.25 + -1a + a2 = 4.25

Factor a perfect square on the left side:
(a + -0.5)(a + -0.5) = 4.25

Calculate the square root of the right side: 2.061552813

Break this problem into two subproblems by setting 
(a + -0.5) equal to 2.061552813 and -2.061552813.

Subproblem 1
a + -0.5 = 2.061552813

Simplifying
a + -0.5 = 2.061552813

Reorder the terms:
-0.5 + a = 2.061552813

Solving
-0.5 + a = 2.061552813

Solving for variable 'a'.

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

Add '0.5' to each side of the equation.
-0.5 + 0.5 + a = 2.061552813 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + a = 2.061552813 + 0.5
a = 2.061552813 + 0.5

Combine like terms: 2.061552813 + 0.5 = 2.561552813
a = 2.561552813

Simplifying
a = 2.561552813

Subproblem 2
a + -0.5 = -2.061552813

Simplifying
a + -0.5 = -2.061552813

Reorder the terms:
-0.5 + a = -2.061552813

Solving
-0.5 + a = -2.061552813

Solving for variable 'a'.

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

Add '0.5' to each side of the equation.
-0.5 + 0.5 + a = -2.061552813 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + a = -2.061552813 + 0.5
a = -2.061552813 + 0.5

Combine like terms: -2.061552813 + 0.5 = -1.561552813
a = -1.561552813

Simplifying
a = -1.561552813

Solution
The solution to the problem is based on the solutions
from the subproblems.
a = {2.561552813, -1.561552813}