10=10k(k-9)

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Solution for 10=10k(k-9) equation: 


Simplifying
10 = 10k(k + -9)

Reorder the terms:
10 = 10k(-9 + k)
10 = (-9 * 10k + k * 10k)
10 = (-90k + 10k2)

Solving
10 = -90k + 10k2

Solving for variable 'k'.

Reorder the terms:
10 + 90k + -10k2 = -90k + 90k + 10k2 + -10k2

Combine like terms: -90k + 90k = 0
10 + 90k + -10k2 = 0 + 10k2 + -10k2
10 + 90k + -10k2 = 10k2 + -10k2

Combine like terms: 10k2 + -10k2 = 0
10 + 90k + -10k2 = 0

Factor out the Greatest Common Factor (GCF), '10'.
10(1 + 9k + -1k2) = 0

Ignore the factor 10.

Subproblem 1
Set the factor '(1 + 9k + -1k2)' equal to zero and attempt to solve:

Simplifying
1 + 9k + -1k2 = 0

Solving
1 + 9k + -1k2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-1 + -9k + k2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -9k + 1 + k2 = 0 + 1

Reorder the terms:
-1 + 1 + -9k + k2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -9k + k2 = 0 + 1
-9k + k2 = 0 + 1

Combine like terms: 0 + 1 = 1
-9k + k2 = 1

The k term is -9k.  Take half its coefficient (-4.5).
Square it (20.25) and add it to both sides.

Add '20.25' to each side of the equation.
-9k + 20.25 + k2 = 1 + 20.25

Reorder the terms:
20.25 + -9k + k2 = 1 + 20.25

Combine like terms: 1 + 20.25 = 21.25
20.25 + -9k + k2 = 21.25

Factor a perfect square on the left side:
(k + -4.5)(k + -4.5) = 21.25

Calculate the square root of the right side: 4.609772229

Break this problem into two subproblems by setting 
(k + -4.5) equal to 4.609772229 and -4.609772229.


Subproblem 1
k + -4.5 = 4.609772229

Simplifying
k + -4.5 = 4.609772229

Reorder the terms:
-4.5 + k = 4.609772229

Solving
-4.5 + k = 4.609772229

Solving for variable 'k'.

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

Add '4.5' to each side of the equation.
-4.5 + 4.5 + k = 4.609772229 + 4.5

Combine like terms: -4.5 + 4.5 = 0.0
0.0 + k = 4.609772229 + 4.5
k = 4.609772229 + 4.5

Combine like terms: 4.609772229 + 4.5 = 9.109772229
k = 9.109772229

Simplifying
k = 9.109772229


Subproblem 2
k + -4.5 = -4.609772229

Simplifying
k + -4.5 = -4.609772229

Reorder the terms:
-4.5 + k = -4.609772229

Solving
-4.5 + k = -4.609772229

Solving for variable 'k'.

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

Add '4.5' to each side of the equation.
-4.5 + 4.5 + k = -4.609772229 + 4.5

Combine like terms: -4.5 + 4.5 = 0.0
0.0 + k = -4.609772229 + 4.5
k = -4.609772229 + 4.5

Combine like terms: -4.609772229 + 4.5 = -0.109772229
k = -0.109772229

Simplifying
k = -0.109772229


Solution
The solution to the problem is based on the solutions
from the subproblems.
k = {9.109772229, -0.109772229}
Solution
k = {9.109772229, -0.109772229}

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