120=3t^2-6t+6

Solution for 120=3t^2-6t+6 equation:

Simplifying
120 = 3t2 + -6t + 6

Reorder the terms:
120 = 6 + -6t + 3t2

Solving
120 = 6 + -6t + 3t2

Solving for variable 't'.

Combine like terms: 120 + -6 = 114
114 + 6t + -3t2 = 6 + -6t + 3t2 + -6 + 6t + -3t2

Reorder the terms:
114 + 6t + -3t2 = 6 + -6 + -6t + 6t + 3t2 + -3t2

Combine like terms: 6 + -6 = 0
114 + 6t + -3t2 = 0 + -6t + 6t + 3t2 + -3t2
114 + 6t + -3t2 = -6t + 6t + 3t2 + -3t2

Combine like terms: -6t + 6t = 0
114 + 6t + -3t2 = 0 + 3t2 + -3t2
114 + 6t + -3t2 = 3t2 + -3t2

Combine like terms: 3t2 + -3t2 = 0
114 + 6t + -3t2 = 0

Factor out the Greatest Common Factor (GCF), '3'.
3(38 + 2t + -1t2) = 0

Ignore the factor 3.
Subproblem 1
Set the factor '(38 + 2t + -1t2)' equal to zero and attempt to solve:

Simplifying
38 + 2t + -1t2 = 0

Solving
38 + 2t + -1t2 = 0

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

Divide each side by '-1'.
-38 + -2t + t2 = 0

Move the constant term to the right:

Add '38' to each side of the equation.
-38 + -2t + 38 + t2 = 0 + 38

Reorder the terms:
-38 + 38 + -2t + t2 = 0 + 38

Combine like terms: -38 + 38 = 0
0 + -2t + t2 = 0 + 38
-2t + t2 = 0 + 38

Combine like terms: 0 + 38 = 38
-2t + t2 = 38

The t term is -2t.  Take half its coefficient (-1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
-2t + 1 + t2 = 38 + 1

Reorder the terms:
1 + -2t + t2 = 38 + 1

Combine like terms: 38 + 1 = 39
1 + -2t + t2 = 39

Factor a perfect square on the left side:
(t + -1)(t + -1) = 39

Calculate the square root of the right side: 6.244997998

Break this problem into two subproblems by setting 
(t + -1) equal to 6.244997998 and -6.244997998.

Subproblem 1
t + -1 = 6.244997998

Simplifying
t + -1 = 6.244997998

Reorder the terms:
-1 + t = 6.244997998

Solving
-1 + t = 6.244997998

Solving for variable 't'.

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

Add '1' to each side of the equation.
-1 + 1 + t = 6.244997998 + 1

Combine like terms: -1 + 1 = 0
0 + t = 6.244997998 + 1
t = 6.244997998 + 1

Combine like terms: 6.244997998 + 1 = 7.244997998
t = 7.244997998

Simplifying
t = 7.244997998

Subproblem 2
t + -1 = -6.244997998

Simplifying
t + -1 = -6.244997998

Reorder the terms:
-1 + t = -6.244997998

Solving
-1 + t = -6.244997998

Solving for variable 't'.

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

Add '1' to each side of the equation.
-1 + 1 + t = -6.244997998 + 1

Combine like terms: -1 + 1 = 0
0 + t = -6.244997998 + 1
t = -6.244997998 + 1

Combine like terms: -6.244997998 + 1 = -5.244997998
t = -5.244997998

Simplifying
t = -5.244997998

Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {7.244997998, -5.244997998}
Solution
t = {7.244997998, -5.244997998}