[k(2k+1)]-[(1+k)(5k-3)]=0

Solution for [k(2k+1)]-[(1+k)(5k-3)]=0 equation:

Simplifying
[k(2k + 1)] + -1[(1 + k)(5k + -3)] = 0

Reorder the terms:
[k(1 + 2k)] + -1[(1 + k)(5k + -3)] = 0
[(1 * k + 2k * k)] + -1[(1 + k)(5k + -3)] = 0
[(1k + 2k2)] + -1[(1 + k)(5k + -3)] = 0
[1k + 2k2] + -1[(1 + k)(5k + -3)] = 0

Remove brackets around [1k + 2k2]
1k + 2k2 + -1[(1 + k)(5k + -3)] = 0

Reorder the terms:
1k + 2k2 + -1[(1 + k)(-3 + 5k)] = 0

Multiply (1 + k) * (-3 + 5k)
1k + 2k2 + -1[(1(-3 + 5k) + k(-3 + 5k))] = 0
1k + 2k2 + -1[((-3 * 1 + 5k * 1) + k(-3 + 5k))] = 0
1k + 2k2 + -1[((-3 + 5k) + k(-3 + 5k))] = 0
1k + 2k2 + -1[(-3 + 5k + (-3 * k + 5k * k))] = 0
1k + 2k2 + -1[(-3 + 5k + (-3k + 5k2))] = 0

Combine like terms: 5k + -3k = 2k
1k + 2k2 + -1[(-3 + 2k + 5k2)] = 0
1k + 2k2 + [-3 * -1 + 2k * -1 + 5k2 * -1] = 0
1k + 2k2 + [3 + -2k + -5k2] = 0

Reorder the terms:
3 + 1k + -2k + 2k2 + -5k2 = 0

Combine like terms: 1k + -2k = -1k
3 + -1k + 2k2 + -5k2 = 0

Combine like terms: 2k2 + -5k2 = -3k2
3 + -1k + -3k2 = 0

Solving
3 + -1k + -3k2 = 0

Solving for variable 'k'.

Begin completing the square.  Divide all terms by
-3 the coefficient of the squared term: 

Divide each side by '-3'.
-1 + 0.3333333333k + k2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + 0.3333333333k + 1 + k2 = 0 + 1

Reorder the terms:
-1 + 1 + 0.3333333333k + k2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + 0.3333333333k + k2 = 0 + 1
0.3333333333k + k2 = 0 + 1

Combine like terms: 0 + 1 = 1
0.3333333333k + k2 = 1

The k term is 0.3333333333k.  Take half its coefficient (0.1666666667).
Square it (0.02777777779) and add it to both sides.

Add '0.02777777779' to each side of the equation.
0.3333333333k + 0.02777777779 + k2 = 1 + 0.02777777779

Reorder the terms:
0.02777777779 + 0.3333333333k + k2 = 1 + 0.02777777779

Combine like terms: 1 + 0.02777777779 = 1.02777777779
0.02777777779 + 0.3333333333k + k2 = 1.02777777779

Factor a perfect square on the left side:
(k + 0.1666666667)(k + 0.1666666667) = 1.02777777779

Calculate the square root of the right side: 1.013793755

Break this problem into two subproblems by setting 
(k + 0.1666666667) equal to 1.013793755 and -1.013793755.

Subproblem 1
k + 0.1666666667 = 1.013793755

Simplifying
k + 0.1666666667 = 1.013793755

Reorder the terms:
0.1666666667 + k = 1.013793755

Solving
0.1666666667 + k = 1.013793755

Solving for variable 'k'.

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

Add '-0.1666666667' to each side of the equation.
0.1666666667 + -0.1666666667 + k = 1.013793755 + -0.1666666667

Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000
0.0000000000 + k = 1.013793755 + -0.1666666667
k = 1.013793755 + -0.1666666667

Combine like terms: 1.013793755 + -0.1666666667 = 0.8471270883
k = 0.8471270883

Simplifying
k = 0.8471270883

Subproblem 2
k + 0.1666666667 = -1.013793755

Simplifying
k + 0.1666666667 = -1.013793755

Reorder the terms:
0.1666666667 + k = -1.013793755

Solving
0.1666666667 + k = -1.013793755

Solving for variable 'k'.

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

Add '-0.1666666667' to each side of the equation.
0.1666666667 + -0.1666666667 + k = -1.013793755 + -0.1666666667

Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000
0.0000000000 + k = -1.013793755 + -0.1666666667
k = -1.013793755 + -0.1666666667

Combine like terms: -1.013793755 + -0.1666666667 = -1.1804604217
k = -1.1804604217

Simplifying
k = -1.1804604217

Solution
The solution to the problem is based on the solutions
from the subproblems.
k = {0.8471270883, -1.1804604217}