2(2k-1)=4k(k-2)

Solution for 2(2k-1)=4k(k-2) equation:

Simplifying
2(2k + -1) = 4k(k + -2)

Reorder the terms:
2(-1 + 2k) = 4k(k + -2)
(-1 * 2 + 2k * 2) = 4k(k + -2)
(-2 + 4k) = 4k(k + -2)

Reorder the terms:
-2 + 4k = 4k(-2 + k)
-2 + 4k = (-2 * 4k + k * 4k)
-2 + 4k = (-8k + 4k2)

Solving
-2 + 4k = -8k + 4k2

Solving for variable 'k'.

Combine like terms: 4k + 8k = 12k
-2 + 12k + -4k2 = -8k + 4k2 + 8k + -4k2

Reorder the terms:
-2 + 12k + -4k2 = -8k + 8k + 4k2 + -4k2

Combine like terms: -8k + 8k = 0
-2 + 12k + -4k2 = 0 + 4k2 + -4k2
-2 + 12k + -4k2 = 4k2 + -4k2

Combine like terms: 4k2 + -4k2 = 0
-2 + 12k + -4k2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-1 + 6k + -2k2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-1 + 6k + -2k2)' equal to zero and attempt to solve:

Simplifying
-1 + 6k + -2k2 = 0

Solving
-1 + 6k + -2k2 = 0

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

Divide each side by '-2'.
0.5 + -3k + k2 = 0

Move the constant term to the right:

Add '-0.5' to each side of the equation.
0.5 + -3k + -0.5 + k2 = 0 + -0.5

Reorder the terms:
0.5 + -0.5 + -3k + k2 = 0 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + -3k + k2 = 0 + -0.5
-3k + k2 = 0 + -0.5

Combine like terms: 0 + -0.5 = -0.5
-3k + k2 = -0.5

The k term is -3k.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3k + 2.25 + k2 = -0.5 + 2.25

Reorder the terms:
2.25 + -3k + k2 = -0.5 + 2.25

Combine like terms: -0.5 + 2.25 = 1.75
2.25 + -3k + k2 = 1.75

Factor a perfect square on the left side:
(k + -1.5)(k + -1.5) = 1.75

Calculate the square root of the right side: 1.322875656

Break this problem into two subproblems by setting 
(k + -1.5) equal to 1.322875656 and -1.322875656.

Subproblem 1
k + -1.5 = 1.322875656

Simplifying
k + -1.5 = 1.322875656

Reorder the terms:
-1.5 + k = 1.322875656

Solving
-1.5 + k = 1.322875656

Solving for variable 'k'.

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

Add '1.5' to each side of the equation.
-1.5 + 1.5 + k = 1.322875656 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + k = 1.322875656 + 1.5
k = 1.322875656 + 1.5

Combine like terms: 1.322875656 + 1.5 = 2.822875656
k = 2.822875656

Simplifying
k = 2.822875656

Subproblem 2
k + -1.5 = -1.322875656

Simplifying
k + -1.5 = -1.322875656

Reorder the terms:
-1.5 + k = -1.322875656

Solving
-1.5 + k = -1.322875656

Solving for variable 'k'.

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

Add '1.5' to each side of the equation.
-1.5 + 1.5 + k = -1.322875656 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + k = -1.322875656 + 1.5
k = -1.322875656 + 1.5

Combine like terms: -1.322875656 + 1.5 = 0.177124344
k = 0.177124344

Simplifying
k = 0.177124344

Solution
The solution to the problem is based on the solutions
from the subproblems.
k = {2.822875656, 0.177124344}
Solution
k = {2.822875656, 0.177124344}