6t^3+t^2+t=0

Solution for 6t^3+t^2+t=0 equation:

Simplifying
6t3 + t2 + t = 0

Reorder the terms:
t + t2 + 6t3 = 0

Solving
t + t2 + 6t3 = 0

Solving for variable 't'.

Factor out the Greatest Common Factor (GCF), 't'.
t(1 + t + 6t2) = 0

Subproblem 1
Set the factor 't' equal to zero and attempt to solve:

Simplifying
t = 0

Solving
t = 0

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

Simplifying
t = 0
Subproblem 2
Set the factor '(1 + t + 6t2)' equal to zero and attempt to solve:

Simplifying
1 + t + 6t2 = 0

Solving
1 + t + 6t2 = 0

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

Divide each side by '6'.
0.1666666667 + 0.1666666667t + t2 = 0

Move the constant term to the right:

Add '-0.1666666667' to each side of the equation.
0.1666666667 + 0.1666666667t + -0.1666666667 + t2 = 0 + -0.1666666667

Reorder the terms:
0.1666666667 + -0.1666666667 + 0.1666666667t + t2 = 0 + -0.1666666667

Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000
0.0000000000 + 0.1666666667t + t2 = 0 + -0.1666666667
0.1666666667t + t2 = 0 + -0.1666666667

Combine like terms: 0 + -0.1666666667 = -0.1666666667
0.1666666667t + t2 = -0.1666666667

The t term is t.  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.
0.1666666667t + 0.25 + t2 = -0.1666666667 + 0.25

Reorder the terms:
0.25 + 0.1666666667t + t2 = -0.1666666667 + 0.25

Combine like terms: -0.1666666667 + 0.25 = 0.0833333333
0.25 + 0.1666666667t + t2 = 0.0833333333

Factor a perfect square on the left side:
(t + 0.5)(t + 0.5) = 0.0833333333

Calculate the square root of the right side: 0.288675135

Break this problem into two subproblems by setting 
(t + 0.5) equal to 0.288675135 and -0.288675135.

Subproblem 1
t + 0.5 = 0.288675135

Simplifying
t + 0.5 = 0.288675135

Reorder the terms:
0.5 + t = 0.288675135

Solving
0.5 + t = 0.288675135

Solving for variable 't'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + t = 0.288675135 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + t = 0.288675135 + -0.5
t = 0.288675135 + -0.5

Combine like terms: 0.288675135 + -0.5 = -0.211324865
t = -0.211324865

Simplifying
t = -0.211324865

Subproblem 2
t + 0.5 = -0.288675135

Simplifying
t + 0.5 = -0.288675135

Reorder the terms:
0.5 + t = -0.288675135

Solving
0.5 + t = -0.288675135

Solving for variable 't'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + t = -0.288675135 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + t = -0.288675135 + -0.5
t = -0.288675135 + -0.5

Combine like terms: -0.288675135 + -0.5 = -0.788675135
t = -0.788675135

Simplifying
t = -0.788675135

Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {-0.211324865, -0.788675135}
Solution
t = {0, -0.211324865, -0.788675135}