24k^2-31k-36=k

Solution for 24k^2-31k-36=k equation:

Simplifying
24k2 + -31k + -36 = k

Reorder the terms:
-36 + -31k + 24k2 = k

Solving
-36 + -31k + 24k2 = k

Solving for variable 'k'.

Reorder the terms:
-36 + -31k + -1k + 24k2 = k + -1k

Combine like terms: -31k + -1k = -32k
-36 + -32k + 24k2 = k + -1k

Combine like terms: k + -1k = 0
-36 + -32k + 24k2 = 0

Factor out the Greatest Common Factor (GCF), '4'.
4(-9 + -8k + 6k2) = 0

Ignore the factor 4.
Subproblem 1
Set the factor '(-9 + -8k + 6k2)' equal to zero and attempt to solve:

Simplifying
-9 + -8k + 6k2 = 0

Solving
-9 + -8k + 6k2 = 0

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

Divide each side by '6'.
-1.5 + -1.333333333k + k2 = 0

Move the constant term to the right:

Add '1.5' to each side of the equation.
-1.5 + -1.333333333k + 1.5 + k2 = 0 + 1.5

Reorder the terms:
-1.5 + 1.5 + -1.333333333k + k2 = 0 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + -1.333333333k + k2 = 0 + 1.5
-1.333333333k + k2 = 0 + 1.5

Combine like terms: 0 + 1.5 = 1.5
-1.333333333k + k2 = 1.5

The k term is -1.333333333k.  Take half its coefficient (-0.6666666665).
Square it (0.4444444442) and add it to both sides.

Add '0.4444444442' to each side of the equation.
-1.333333333k + 0.4444444442 + k2 = 1.5 + 0.4444444442

Reorder the terms:
0.4444444442 + -1.333333333k + k2 = 1.5 + 0.4444444442

Combine like terms: 1.5 + 0.4444444442 = 1.9444444442
0.4444444442 + -1.333333333k + k2 = 1.9444444442

Factor a perfect square on the left side:
(k + -0.6666666665)(k + -0.6666666665) = 1.9444444442

Calculate the square root of the right side: 1.394433377

Break this problem into two subproblems by setting 
(k + -0.6666666665) equal to 1.394433377 and -1.394433377.

Subproblem 1
k + -0.6666666665 = 1.394433377

Simplifying
k + -0.6666666665 = 1.394433377

Reorder the terms:
-0.6666666665 + k = 1.394433377

Solving
-0.6666666665 + k = 1.394433377

Solving for variable 'k'.

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

Add '0.6666666665' to each side of the equation.
-0.6666666665 + 0.6666666665 + k = 1.394433377 + 0.6666666665

Combine like terms: -0.6666666665 + 0.6666666665 = 0.0000000000
0.0000000000 + k = 1.394433377 + 0.6666666665
k = 1.394433377 + 0.6666666665

Combine like terms: 1.394433377 + 0.6666666665 = 2.0611000435
k = 2.0611000435

Simplifying
k = 2.0611000435

Subproblem 2
k + -0.6666666665 = -1.394433377

Simplifying
k + -0.6666666665 = -1.394433377

Reorder the terms:
-0.6666666665 + k = -1.394433377

Solving
-0.6666666665 + k = -1.394433377

Solving for variable 'k'.

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

Add '0.6666666665' to each side of the equation.
-0.6666666665 + 0.6666666665 + k = -1.394433377 + 0.6666666665

Combine like terms: -0.6666666665 + 0.6666666665 = 0.0000000000
0.0000000000 + k = -1.394433377 + 0.6666666665
k = -1.394433377 + 0.6666666665

Combine like terms: -1.394433377 + 0.6666666665 = -0.7277667105
k = -0.7277667105

Simplifying
k = -0.7277667105

Solution
The solution to the problem is based on the solutions
from the subproblems.
k = {2.0611000435, -0.7277667105}
Solution
k = {2.0611000435, -0.7277667105}