k(k+1)-253.2=0

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Solution for k(k+1)-253.2=0 equation: 


Simplifying
k(k + 1) + -253.2 = 0

Reorder the terms:
k(1 + k) + -253.2 = 0
(1 * k + k * k) + -253.2 = 0
(1k + k2) + -253.2 = 0

Reorder the terms:
-253.2 + 1k + k2 = 0

Solving
-253.2 + 1k + k2 = 0

Solving for variable 'k'.

Begin completing the square.

Move the constant term to the right:

Add '253.2' to each side of the equation.
-253.2 + 1k + 253.2 + k2 = 0 + 253.2

Reorder the terms:
-253.2 + 253.2 + 1k + k2 = 0 + 253.2

Combine like terms: -253.2 + 253.2 = 0.0
0.0 + 1k + k2 = 0 + 253.2
1k + k2 = 0 + 253.2

Combine like terms: 0 + 253.2 = 253.2
1k + k2 = 253.2

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 = 253.2 + 0.25

Reorder the terms:
0.25 + k + k2 = 253.2 + 0.25

Combine like terms: 253.2 + 0.25 = 253.45
0.25 + k + k2 = 253.45

Factor a perfect square on the left side:
(k + 0.5)(k + 0.5) = 253.45

Calculate the square root of the right side: 15.920113065

Break this problem into two subproblems by setting 
(k + 0.5) equal to 15.920113065 and -15.920113065.


Subproblem 1
k + 0.5 = 15.920113065

Simplifying
k + 0.5 = 15.920113065

Reorder the terms:
0.5 + k = 15.920113065

Solving
0.5 + k = 15.920113065

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 = 15.920113065 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + k = 15.920113065 + -0.5
k = 15.920113065 + -0.5

Combine like terms: 15.920113065 + -0.5 = 15.420113065
k = 15.420113065

Simplifying
k = 15.420113065


Subproblem 2
k + 0.5 = -15.920113065

Simplifying
k + 0.5 = -15.920113065

Reorder the terms:
0.5 + k = -15.920113065

Solving
0.5 + k = -15.920113065

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 = -15.920113065 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + k = -15.920113065 + -0.5
k = -15.920113065 + -0.5

Combine like terms: -15.920113065 + -0.5 = -16.420113065
k = -16.420113065

Simplifying
k = -16.420113065


Solution
The solution to the problem is based on the solutions
from the subproblems.
k = {15.420113065, -16.420113065}

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