3t^2-6t-27=0

Solution for 3t^2-6t-27=0 equation:

Simplifying
3t2 + -6t + -27 = 0

Reorder the terms:
-27 + -6t + 3t2 = 0

Solving
-27 + -6t + 3t2 = 0

Solving for variable 't'.

Factor out the Greatest Common Factor (GCF), '3'.
3(-9 + -2t + t2) = 0

Ignore the factor 3.
Subproblem 1
Set the factor '(-9 + -2t + t2)' equal to zero and attempt to solve:

Simplifying
-9 + -2t + t2 = 0

Solving
-9 + -2t + t2 = 0

Begin completing the square.

Move the constant term to the right:

Add '9' to each side of the equation.
-9 + -2t + 9 + t2 = 0 + 9

Reorder the terms:
-9 + 9 + -2t + t2 = 0 + 9

Combine like terms: -9 + 9 = 0
0 + -2t + t2 = 0 + 9
-2t + t2 = 0 + 9

Combine like terms: 0 + 9 = 9
-2t + t2 = 9

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 = 9 + 1

Reorder the terms:
1 + -2t + t2 = 9 + 1

Combine like terms: 9 + 1 = 10
1 + -2t + t2 = 10

Factor a perfect square on the left side:
(t + -1)(t + -1) = 10

Calculate the square root of the right side: 3.16227766

Break this problem into two subproblems by setting 
(t + -1) equal to 3.16227766 and -3.16227766.

Subproblem 1
t + -1 = 3.16227766

Simplifying
t + -1 = 3.16227766

Reorder the terms:
-1 + t = 3.16227766

Solving
-1 + t = 3.16227766

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 = 3.16227766 + 1

Combine like terms: -1 + 1 = 0
0 + t = 3.16227766 + 1
t = 3.16227766 + 1

Combine like terms: 3.16227766 + 1 = 4.16227766
t = 4.16227766

Simplifying
t = 4.16227766

Subproblem 2
t + -1 = -3.16227766

Simplifying
t + -1 = -3.16227766

Reorder the terms:
-1 + t = -3.16227766

Solving
-1 + t = -3.16227766

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 = -3.16227766 + 1

Combine like terms: -1 + 1 = 0
0 + t = -3.16227766 + 1
t = -3.16227766 + 1

Combine like terms: -3.16227766 + 1 = -2.16227766
t = -2.16227766

Simplifying
t = -2.16227766

Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {4.16227766, -2.16227766}
Solution
t = {4.16227766, -2.16227766}