2k[k+1]+2=10

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Solution for 2k[k+1]+2=10 equation: 


Simplifying
2k[k + 1] + 2 = 10

Reorder the terms:
2k[1 + k] + 2 = 10
[1 * 2k + k * 2k] + 2 = 10
[2k + 2k2] + 2 = 10

Reorder the terms:
2 + 2k + 2k2 = 10

Solving
2 + 2k + 2k2 = 10

Solving for variable 'k'.

Reorder the terms:
2 + -10 + 2k + 2k2 = 10 + -10

Combine like terms: 2 + -10 = -8
-8 + 2k + 2k2 = 10 + -10

Combine like terms: 10 + -10 = 0
-8 + 2k + 2k2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-4 + k + k2) = 0

Ignore the factor 2.

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

Simplifying
-4 + k + k2 = 0

Solving
-4 + k + k2 = 0

Begin completing the square.

Move the constant term to the right:

Add '4' to each side of the equation.
-4 + k + 4 + k2 = 0 + 4

Reorder the terms:
-4 + 4 + k + k2 = 0 + 4

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

Combine like terms: 0 + 4 = 4
k + k2 = 4

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.
k + 0.25 + k2 = 4 + 0.25

Reorder the terms:
0.25 + k + k2 = 4 + 0.25

Combine like terms: 4 + 0.25 = 4.25
0.25 + k + k2 = 4.25

Factor a perfect square on the left side:
(k + 0.5)(k + 0.5) = 4.25

Calculate the square root of the right side: 2.061552813

Break this problem into two subproblems by setting 
(k + 0.5) equal to 2.061552813 and -2.061552813.


Subproblem 1
k + 0.5 = 2.061552813

Simplifying
k + 0.5 = 2.061552813

Reorder the terms:
0.5 + k = 2.061552813

Solving
0.5 + k = 2.061552813

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 = 2.061552813 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + k = 2.061552813 + -0.5
k = 2.061552813 + -0.5

Combine like terms: 2.061552813 + -0.5 = 1.561552813
k = 1.561552813

Simplifying
k = 1.561552813


Subproblem 2
k + 0.5 = -2.061552813

Simplifying
k + 0.5 = -2.061552813

Reorder the terms:
0.5 + k = -2.061552813

Solving
0.5 + k = -2.061552813

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 = -2.061552813 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + k = -2.061552813 + -0.5
k = -2.061552813 + -0.5

Combine like terms: -2.061552813 + -0.5 = -2.561552813
k = -2.561552813

Simplifying
k = -2.561552813


Solution
The solution to the problem is based on the solutions
from the subproblems.
k = {1.561552813, -2.561552813}
Solution
k = {1.561552813, -2.561552813}

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