(k-6)(k-1)=k-2

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Solution for (k-6)(k-1)=k-2 equation: 


Simplifying
(k + -6)(k + -1) = k + -2

Reorder the terms:
(-6 + k)(k + -1) = k + -2

Reorder the terms:
(-6 + k)(-1 + k) = k + -2

Multiply (-6 + k) * (-1 + k)
(-6(-1 + k) + k(-1 + k)) = k + -2
((-1 * -6 + k * -6) + k(-1 + k)) = k + -2
((6 + -6k) + k(-1 + k)) = k + -2
(6 + -6k + (-1 * k + k * k)) = k + -2
(6 + -6k + (-1k + k2)) = k + -2

Combine like terms: -6k + -1k = -7k
(6 + -7k + k2) = k + -2

Reorder the terms:
6 + -7k + k2 = -2 + k

Solving
6 + -7k + k2 = -2 + k

Solving for variable 'k'.

Reorder the terms:
6 + 2 + -7k + -1k + k2 = -2 + k + 2 + -1k

Combine like terms: 6 + 2 = 8
8 + -7k + -1k + k2 = -2 + k + 2 + -1k

Combine like terms: -7k + -1k = -8k
8 + -8k + k2 = -2 + k + 2 + -1k

Reorder the terms:
8 + -8k + k2 = -2 + 2 + k + -1k

Combine like terms: -2 + 2 = 0
8 + -8k + k2 = 0 + k + -1k
8 + -8k + k2 = k + -1k

Combine like terms: k + -1k = 0
8 + -8k + k2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-8' to each side of the equation.
8 + -8k + -8 + k2 = 0 + -8

Reorder the terms:
8 + -8 + -8k + k2 = 0 + -8

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

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

The k term is -8k.  Take half its coefficient (-4).
Square it (16) and add it to both sides.

Add '16' to each side of the equation.
-8k + 16 + k2 = -8 + 16

Reorder the terms:
16 + -8k + k2 = -8 + 16

Combine like terms: -8 + 16 = 8
16 + -8k + k2 = 8

Factor a perfect square on the left side:
(k + -4)(k + -4) = 8

Calculate the square root of the right side: 2.828427125

Break this problem into two subproblems by setting 
(k + -4) equal to 2.828427125 and -2.828427125.


Subproblem 1
k + -4 = 2.828427125

Simplifying
k + -4 = 2.828427125

Reorder the terms:
-4 + k = 2.828427125

Solving
-4 + k = 2.828427125

Solving for variable 'k'.

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

Add '4' to each side of the equation.
-4 + 4 + k = 2.828427125 + 4

Combine like terms: -4 + 4 = 0
0 + k = 2.828427125 + 4
k = 2.828427125 + 4

Combine like terms: 2.828427125 + 4 = 6.828427125
k = 6.828427125

Simplifying
k = 6.828427125


Subproblem 2
k + -4 = -2.828427125

Simplifying
k + -4 = -2.828427125

Reorder the terms:
-4 + k = -2.828427125

Solving
-4 + k = -2.828427125

Solving for variable 'k'.

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

Add '4' to each side of the equation.
-4 + 4 + k = -2.828427125 + 4

Combine like terms: -4 + 4 = 0
0 + k = -2.828427125 + 4
k = -2.828427125 + 4

Combine like terms: -2.828427125 + 4 = 1.171572875
k = 1.171572875

Simplifying
k = 1.171572875


Solution
The solution to the problem is based on the solutions
from the subproblems.
k = {6.828427125, 1.171572875}

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