6t^2+81t+12=0

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Solution for 6t^2+81t+12=0 equation: 


Simplifying
6t2 + 81t + 12 = 0

Reorder the terms:
12 + 81t + 6t2 = 0

Solving
12 + 81t + 6t2 = 0

Solving for variable 't'.

Factor out the Greatest Common Factor (GCF), '3'.
3(4 + 27t + 2t2) = 0

Ignore the factor 3.

Subproblem 1
Set the factor '(4 + 27t + 2t2)' equal to zero and attempt to solve:

Simplifying
4 + 27t + 2t2 = 0

Solving
4 + 27t + 2t2 = 0

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

Divide each side by '2'.
2 + 13.5t + t2 = 0

Move the constant term to the right:

Add '-2' to each side of the equation.
2 + 13.5t + -2 + t2 = 0 + -2

Reorder the terms:
2 + -2 + 13.5t + t2 = 0 + -2

Combine like terms: 2 + -2 = 0
0 + 13.5t + t2 = 0 + -2
13.5t + t2 = 0 + -2

Combine like terms: 0 + -2 = -2
13.5t + t2 = -2

The t term is 13.5t.  Take half its coefficient (6.75).
Square it (45.5625) and add it to both sides.

Add '45.5625' to each side of the equation.
13.5t + 45.5625 + t2 = -2 + 45.5625

Reorder the terms:
45.5625 + 13.5t + t2 = -2 + 45.5625

Combine like terms: -2 + 45.5625 = 43.5625
45.5625 + 13.5t + t2 = 43.5625

Factor a perfect square on the left side:
(t + 6.75)(t + 6.75) = 43.5625

Calculate the square root of the right side: 6.600189391

Break this problem into two subproblems by setting 
(t + 6.75) equal to 6.600189391 and -6.600189391.


Subproblem 1
t + 6.75 = 6.600189391

Simplifying
t + 6.75 = 6.600189391

Reorder the terms:
6.75 + t = 6.600189391

Solving
6.75 + t = 6.600189391

Solving for variable 't'.

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

Add '-6.75' to each side of the equation.
6.75 + -6.75 + t = 6.600189391 + -6.75

Combine like terms: 6.75 + -6.75 = 0.00
0.00 + t = 6.600189391 + -6.75
t = 6.600189391 + -6.75

Combine like terms: 6.600189391 + -6.75 = -0.149810609
t = -0.149810609

Simplifying
t = -0.149810609


Subproblem 2
t + 6.75 = -6.600189391

Simplifying
t + 6.75 = -6.600189391

Reorder the terms:
6.75 + t = -6.600189391

Solving
6.75 + t = -6.600189391

Solving for variable 't'.

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

Add '-6.75' to each side of the equation.
6.75 + -6.75 + t = -6.600189391 + -6.75

Combine like terms: 6.75 + -6.75 = 0.00
0.00 + t = -6.600189391 + -6.75
t = -6.600189391 + -6.75

Combine like terms: -6.600189391 + -6.75 = -13.350189391
t = -13.350189391

Simplifying
t = -13.350189391


Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {-0.149810609, -13.350189391}
Solution
t = {-0.149810609, -13.350189391}

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