(2k+1)(2k+1)=8

Solution for (2k+1)(2k+1)=8 equation:

Simplifying
(2k + 1)(2k + 1) = 8

Reorder the terms:
(1 + 2k)(2k + 1) = 8

Reorder the terms:
(1 + 2k)(1 + 2k) = 8

Multiply (1 + 2k) * (1 + 2k)
(1(1 + 2k) + 2k * (1 + 2k)) = 8
((1 * 1 + 2k * 1) + 2k * (1 + 2k)) = 8
((1 + 2k) + 2k * (1 + 2k)) = 8
(1 + 2k + (1 * 2k + 2k * 2k)) = 8
(1 + 2k + (2k + 4k2)) = 8

Combine like terms: 2k + 2k = 4k
(1 + 4k + 4k2) = 8

Solving
1 + 4k + 4k2 = 8

Solving for variable 'k'.

Reorder the terms:
1 + -8 + 4k + 4k2 = 8 + -8

Combine like terms: 1 + -8 = -7
-7 + 4k + 4k2 = 8 + -8

Combine like terms: 8 + -8 = 0
-7 + 4k + 4k2 = 0

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

Divide each side by '4'.
-1.75 + k + k2 = 0

Move the constant term to the right:

Add '1.75' to each side of the equation.
-1.75 + k + 1.75 + k2 = 0 + 1.75

Reorder the terms:
-1.75 + 1.75 + k + k2 = 0 + 1.75

Combine like terms: -1.75 + 1.75 = 0.00
0.00 + k + k2 = 0 + 1.75
k + k2 = 0 + 1.75

Combine like terms: 0 + 1.75 = 1.75
k + k2 = 1.75

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 = 1.75 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + k2 = 1.75 + 0.25

Combine like terms: 1.75 + 0.25 = 2
1.25 + k2 = 2

Factor a perfect square on the left side:
(k + 0.5)(k + 0.5) = 2

Calculate the square root of the right side: 1.414213562

Break this problem into two subproblems by setting 
(k + 0.5) equal to 1.414213562 and -1.414213562.

Subproblem 1
k + 0.5 = 1.414213562

Simplifying
k + 0.5 = 1.414213562

Reorder the terms:
0.5 + k = 1.414213562

Solving
0.5 + k = 1.414213562

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 = 1.414213562 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + k = 1.414213562 + -0.5
k = 1.414213562 + -0.5

Combine like terms: 1.414213562 + -0.5 = 0.914213562
k = 0.914213562

Simplifying
k = 0.914213562

Subproblem 2
k + 0.5 = -1.414213562

Simplifying
k + 0.5 = -1.414213562

Reorder the terms:
0.5 + k = -1.414213562

Solving
0.5 + k = -1.414213562

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 = -1.414213562 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + k = -1.414213562 + -0.5
k = -1.414213562 + -0.5

Combine like terms: -1.414213562 + -0.5 = -1.914213562
k = -1.914213562

Simplifying
k = -1.914213562

Solution
The solution to the problem is based on the solutions
from the subproblems.
k = {0.914213562, -1.914213562}