3k^2+9k+1=0

Solution for 3k^2+9k+1=0 equation:

Simplifying
3k2 + 9k + 1 = 0

Reorder the terms:
1 + 9k + 3k2 = 0

Solving
1 + 9k + 3k2 = 0

Solving for variable 'k'.

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

Divide each side by '3'.
0.3333333333 + 3k + k2 = 0

Move the constant term to the right:

Add '-0.3333333333' to each side of the equation.
0.3333333333 + 3k + -0.3333333333 + k2 = 0 + -0.3333333333

Reorder the terms:
0.3333333333 + -0.3333333333 + 3k + k2 = 0 + -0.3333333333

Combine like terms: 0.3333333333 + -0.3333333333 = 0.0000000000
0.0000000000 + 3k + k2 = 0 + -0.3333333333
3k + k2 = 0 + -0.3333333333

Combine like terms: 0 + -0.3333333333 = -0.3333333333
3k + k2 = -0.3333333333

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 = -0.3333333333 + 2.25

Reorder the terms:
2.25 + 3k + k2 = -0.3333333333 + 2.25

Combine like terms: -0.3333333333 + 2.25 = 1.9166666667
2.25 + 3k + k2 = 1.9166666667

Factor a perfect square on the left side:
(k + 1.5)(k + 1.5) = 1.9166666667

Calculate the square root of the right side: 1.38443731

Break this problem into two subproblems by setting 
(k + 1.5) equal to 1.38443731 and -1.38443731.

Subproblem 1
k + 1.5 = 1.38443731

Simplifying
k + 1.5 = 1.38443731

Reorder the terms:
1.5 + k = 1.38443731

Solving
1.5 + k = 1.38443731

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 = 1.38443731 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + k = 1.38443731 + -1.5
k = 1.38443731 + -1.5

Combine like terms: 1.38443731 + -1.5 = -0.11556269
k = -0.11556269

Simplifying
k = -0.11556269

Subproblem 2
k + 1.5 = -1.38443731

Simplifying
k + 1.5 = -1.38443731

Reorder the terms:
1.5 + k = -1.38443731

Solving
1.5 + k = -1.38443731

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 = -1.38443731 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + k = -1.38443731 + -1.5
k = -1.38443731 + -1.5

Combine like terms: -1.38443731 + -1.5 = -2.88443731
k = -2.88443731

Simplifying
k = -2.88443731

Solution
The solution to the problem is based on the solutions
from the subproblems.
k = {-0.11556269, -2.88443731}