4k(k+1)=1

Solution for 4k(k+1)=1 equation:

Simplifying
4k(k + 1) = 1

Reorder the terms:
4k(1 + k) = 1
(1 * 4k + k * 4k) = 1
(4k + 4k2) = 1

Solving
4k + 4k2 = 1

Solving for variable 'k'.

Reorder the terms:
-1 + 4k + 4k2 = 1 + -1

Combine like terms: 1 + -1 = 0
-1 + 4k + 4k2 = 0

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

Divide each side by '4'.
-0.25 + k + k2 = 0

Move the constant term to the right:

Add '0.25' to each side of the equation.
-0.25 + k + 0.25 + k2 = 0 + 0.25

Reorder the terms:
-0.25 + 0.25 + k + k2 = 0 + 0.25

Combine like terms: -0.25 + 0.25 = 0.00
0.00 + k + k2 = 0 + 0.25
k + k2 = 0 + 0.25

Combine like terms: 0 + 0.25 = 0.25
k + k2 = 0.25

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 = 0.25 + 0.25

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

Combine like terms: 0.25 + 0.25 = 0.5
1.25 + k2 = 0.5

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

Calculate the square root of the right side: 0.707106781

Break this problem into two subproblems by setting 
(k + 0.5) equal to 0.707106781 and -0.707106781.

Subproblem 1
k + 0.5 = 0.707106781

Simplifying
k + 0.5 = 0.707106781

Reorder the terms:
0.5 + k = 0.707106781

Solving
0.5 + k = 0.707106781

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 = 0.707106781 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + k = 0.707106781 + -0.5
k = 0.707106781 + -0.5

Combine like terms: 0.707106781 + -0.5 = 0.207106781
k = 0.207106781

Simplifying
k = 0.207106781

Subproblem 2
k + 0.5 = -0.707106781

Simplifying
k + 0.5 = -0.707106781

Reorder the terms:
0.5 + k = -0.707106781

Solving
0.5 + k = -0.707106781

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 = -0.707106781 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + k = -0.707106781 + -0.5
k = -0.707106781 + -0.5

Combine like terms: -0.707106781 + -0.5 = -1.207106781
k = -1.207106781

Simplifying
k = -1.207106781

Solution
The solution to the problem is based on the solutions
from the subproblems.
k = {0.207106781, -1.207106781}