0.5k(k+1)=1

Solution for 0.5k(k+1)=1 equation:

Simplifying
0.5k(k + 1) = 1

Reorder the terms:
0.5k(1 + k) = 1
(1 * 0.5k + k * 0.5k) = 1
(0.5k + 0.5k2) = 1

Solving
0.5k + 0.5k2 = 1

Solving for variable 'k'.

Reorder the terms:
-1 + 0.5k + 0.5k2 = 1 + -1

Combine like terms: 1 + -1 = 0
-1 + 0.5k + 0.5k2 = 0

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

Divide each side by '0.5'.
-2 + k + k2 = 0

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + k + 2 + k2 = 0 + 2

Reorder the terms:
-2 + 2 + k + k2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + k + k2 = 0 + 2
k + k2 = 0 + 2

Combine like terms: 0 + 2 = 2
k + k2 = 2

The k term is k.  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 + k2 = 2 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + k2 = 2 + 0.25

Combine like terms: 2 + 0.25 = 2.25
1.25 + k2 = 2.25

Factor a perfect square on the left side:
(k + 0.5)(k + 0.5) = 2.25

Calculate the square root of the right side: 1.5

Break this problem into two subproblems by setting 
(k + 0.5) equal to 1.5 and -1.5.

Subproblem 1
k + 0.5 = 1.5

Simplifying
k + 0.5 = 1.5

Reorder the terms:
0.5 + k = 1.5

Solving
0.5 + k = 1.5

Solving for variable 'k'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + k = 1.5 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + k = 1.5 + -0.5
k = 1.5 + -0.5

Combine like terms: 1.5 + -0.5 = 1
k = 1

Simplifying
k = 1

Subproblem 2
k + 0.5 = -1.5

Simplifying
k + 0.5 = -1.5

Reorder the terms:
0.5 + k = -1.5

Solving
0.5 + k = -1.5

Solving for variable 'k'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + k = -1.5 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + k = -1.5 + -0.5
k = -1.5 + -0.5

Combine like terms: -1.5 + -0.5 = -2
k = -2

Simplifying
k = -2

Solution
The solution to the problem is based on the solutions
from the subproblems.
k = {1, -2}