3t^2+t-6=0

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

Simplifying
3t2 + t + -6 = 0

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

Solving
-6 + t + 3t2 = 0

Solving for variable 't'.

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

Divide each side by '3'.
-2 + 0.3333333333t + t2 = 0

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + 0.3333333333t + 2 + t2 = 0 + 2

Reorder the terms:
-2 + 2 + 0.3333333333t + t2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + 0.3333333333t + t2 = 0 + 2
0.3333333333t + t2 = 0 + 2

Combine like terms: 0 + 2 = 2
0.3333333333t + t2 = 2

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.
0.3333333333t + 0.25 + t2 = 2 + 0.25

Reorder the terms:
0.25 + 0.3333333333t + t2 = 2 + 0.25

Combine like terms: 2 + 0.25 = 2.25
0.25 + 0.3333333333t + t2 = 2.25

Factor a perfect square on the left side:
(t + 0.5)(t + 0.5) = 2.25

Calculate the square root of the right side: 1.5

Break this problem into two subproblems by setting 
(t + 0.5) equal to 1.5 and -1.5.

Subproblem 1
t + 0.5 = 1.5

Simplifying
t + 0.5 = 1.5

Reorder the terms:
0.5 + t = 1.5

Solving
0.5 + t = 1.5

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 = 1.5 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + t = 1.5 + -0.5
t = 1.5 + -0.5

Combine like terms: 1.5 + -0.5 = 1
t = 1

Simplifying
t = 1

Subproblem 2
t + 0.5 = -1.5

Simplifying
t + 0.5 = -1.5

Reorder the terms:
0.5 + t = -1.5

Solving
0.5 + t = -1.5

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 = -1.5 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + t = -1.5 + -0.5
t = -1.5 + -0.5

Combine like terms: -1.5 + -0.5 = -2
t = -2

Simplifying
t = -2

Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {1, -2}