k(k+1)-525.6=0

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Solution for k(k+1)-525.6=0 equation: 


Simplifying
k(k + 1) + -525.6 = 0

Reorder the terms:
k(1 + k) + -525.6 = 0
(1 * k + k * k) + -525.6 = 0
(1k + k2) + -525.6 = 0

Reorder the terms:
-525.6 + 1k + k2 = 0

Solving
-525.6 + 1k + k2 = 0

Solving for variable 'k'.

Begin completing the square.

Move the constant term to the right:

Add '525.6' to each side of the equation.
-525.6 + 1k + 525.6 + k2 = 0 + 525.6

Reorder the terms:
-525.6 + 525.6 + 1k + k2 = 0 + 525.6

Combine like terms: -525.6 + 525.6 = 0.0
0.0 + 1k + k2 = 0 + 525.6
1k + k2 = 0 + 525.6

Combine like terms: 0 + 525.6 = 525.6
1k + k2 = 525.6

The k term is 1k.  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.
k + 0.25 + k2 = 525.6 + 0.25

Reorder the terms:
0.25 + k + k2 = 525.6 + 0.25

Combine like terms: 525.6 + 0.25 = 525.85
0.25 + k + k2 = 525.85

Factor a perfect square on the left side:
(k + 0.5)(k + 0.5) = 525.85

Calculate the square root of the right side: 22.931419494

Break this problem into two subproblems by setting 
(k + 0.5) equal to 22.931419494 and -22.931419494.


Subproblem 1
k + 0.5 = 22.931419494

Simplifying
k + 0.5 = 22.931419494

Reorder the terms:
0.5 + k = 22.931419494

Solving
0.5 + k = 22.931419494

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 = 22.931419494 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + k = 22.931419494 + -0.5
k = 22.931419494 + -0.5

Combine like terms: 22.931419494 + -0.5 = 22.431419494
k = 22.431419494

Simplifying
k = 22.431419494


Subproblem 2
k + 0.5 = -22.931419494

Simplifying
k + 0.5 = -22.931419494

Reorder the terms:
0.5 + k = -22.931419494

Solving
0.5 + k = -22.931419494

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 = -22.931419494 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + k = -22.931419494 + -0.5
k = -22.931419494 + -0.5

Combine like terms: -22.931419494 + -0.5 = -23.431419494
k = -23.431419494

Simplifying
k = -23.431419494


Solution
The solution to the problem is based on the solutions
from the subproblems.
k = {22.431419494, -23.431419494}

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