(2k+1)(k+1)=2k+1(13)

Solution for (2k+1)(k+1)=2k+1(13) equation:

Simplifying
(2k + 1)(k + 1) = 2k + 1(13)

Reorder the terms:
(1 + 2k)(k + 1) = 2k + 1(13)

Reorder the terms:
(1 + 2k)(1 + k) = 2k + 1(13)

Multiply (1 + 2k) * (1 + k)
(1(1 + k) + 2k * (1 + k)) = 2k + 1(13)
((1 * 1 + k * 1) + 2k * (1 + k)) = 2k + 1(13)
((1 + 1k) + 2k * (1 + k)) = 2k + 1(13)
(1 + 1k + (1 * 2k + k * 2k)) = 2k + 1(13)
(1 + 1k + (2k + 2k2)) = 2k + 1(13)

Combine like terms: 1k + 2k = 3k
(1 + 3k + 2k2) = 2k + 1(13)

Multiply 1 * 13
1 + 3k + 2k2 = 2k + 13

Reorder the terms:
1 + 3k + 2k2 = 13 + 2k

Solving
1 + 3k + 2k2 = 13 + 2k

Solving for variable 'k'.

Reorder the terms:
1 + -13 + 3k + -2k + 2k2 = 13 + 2k + -13 + -2k

Combine like terms: 1 + -13 = -12
-12 + 3k + -2k + 2k2 = 13 + 2k + -13 + -2k

Combine like terms: 3k + -2k = 1k
-12 + 1k + 2k2 = 13 + 2k + -13 + -2k

Reorder the terms:
-12 + 1k + 2k2 = 13 + -13 + 2k + -2k

Combine like terms: 13 + -13 = 0
-12 + 1k + 2k2 = 0 + 2k + -2k
-12 + 1k + 2k2 = 2k + -2k

Combine like terms: 2k + -2k = 0
-12 + 1k + 2k2 = 0

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

Divide each side by '2'.
-6 + 0.5k + k2 = 0

Move the constant term to the right:

Add '6' to each side of the equation.
-6 + 0.5k + 6 + k2 = 0 + 6

Reorder the terms:
-6 + 6 + 0.5k + k2 = 0 + 6

Combine like terms: -6 + 6 = 0
0 + 0.5k + k2 = 0 + 6
0.5k + k2 = 0 + 6

Combine like terms: 0 + 6 = 6
0.5k + k2 = 6

The k term is 0.5k.  Take half its coefficient (0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
0.5k + 0.0625 + k2 = 6 + 0.0625

Reorder the terms:
0.0625 + 0.5k + k2 = 6 + 0.0625

Combine like terms: 6 + 0.0625 = 6.0625
0.0625 + 0.5k + k2 = 6.0625

Factor a perfect square on the left side:
(k + 0.25)(k + 0.25) = 6.0625

Calculate the square root of the right side: 2.46221445

Break this problem into two subproblems by setting 
(k + 0.25) equal to 2.46221445 and -2.46221445.

Subproblem 1
k + 0.25 = 2.46221445

Simplifying
k + 0.25 = 2.46221445

Reorder the terms:
0.25 + k = 2.46221445

Solving
0.25 + k = 2.46221445

Solving for variable 'k'.

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

Add '-0.25' to each side of the equation.
0.25 + -0.25 + k = 2.46221445 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + k = 2.46221445 + -0.25
k = 2.46221445 + -0.25

Combine like terms: 2.46221445 + -0.25 = 2.21221445
k = 2.21221445

Simplifying
k = 2.21221445

Subproblem 2
k + 0.25 = -2.46221445

Simplifying
k + 0.25 = -2.46221445

Reorder the terms:
0.25 + k = -2.46221445

Solving
0.25 + k = -2.46221445

Solving for variable 'k'.

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

Add '-0.25' to each side of the equation.
0.25 + -0.25 + k = -2.46221445 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + k = -2.46221445 + -0.25
k = -2.46221445 + -0.25

Combine like terms: -2.46221445 + -0.25 = -2.71221445
k = -2.71221445

Simplifying
k = -2.71221445

Solution
The solution to the problem is based on the solutions
from the subproblems.
k = {2.21221445, -2.71221445}