3t^2-81t-24=0

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Solution for 3t^2-81t-24=0 equation: 


Simplifying
3t2 + -81t + -24 = 0

Reorder the terms:
-24 + -81t + 3t2 = 0

Solving
-24 + -81t + 3t2 = 0

Solving for variable 't'.

Factor out the Greatest Common Factor (GCF), '3'.
3(-8 + -27t + t2) = 0

Ignore the factor 3.

Subproblem 1
Set the factor '(-8 + -27t + t2)' equal to zero and attempt to solve:

Simplifying
-8 + -27t + t2 = 0

Solving
-8 + -27t + t2 = 0

Begin completing the square.

Move the constant term to the right:

Add '8' to each side of the equation.
-8 + -27t + 8 + t2 = 0 + 8

Reorder the terms:
-8 + 8 + -27t + t2 = 0 + 8

Combine like terms: -8 + 8 = 0
0 + -27t + t2 = 0 + 8
-27t + t2 = 0 + 8

Combine like terms: 0 + 8 = 8
-27t + t2 = 8

The t term is -27t.  Take half its coefficient (-13.5).
Square it (182.25) and add it to both sides.

Add '182.25' to each side of the equation.
-27t + 182.25 + t2 = 8 + 182.25

Reorder the terms:
182.25 + -27t + t2 = 8 + 182.25

Combine like terms: 8 + 182.25 = 190.25
182.25 + -27t + t2 = 190.25

Factor a perfect square on the left side:
(t + -13.5)(t + -13.5) = 190.25

Calculate the square root of the right side: 13.793114224

Break this problem into two subproblems by setting 
(t + -13.5) equal to 13.793114224 and -13.793114224.


Subproblem 1
t + -13.5 = 13.793114224

Simplifying
t + -13.5 = 13.793114224

Reorder the terms:
-13.5 + t = 13.793114224

Solving
-13.5 + t = 13.793114224

Solving for variable 't'.

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

Add '13.5' to each side of the equation.
-13.5 + 13.5 + t = 13.793114224 + 13.5

Combine like terms: -13.5 + 13.5 = 0.0
0.0 + t = 13.793114224 + 13.5
t = 13.793114224 + 13.5

Combine like terms: 13.793114224 + 13.5 = 27.293114224
t = 27.293114224

Simplifying
t = 27.293114224


Subproblem 2
t + -13.5 = -13.793114224

Simplifying
t + -13.5 = -13.793114224

Reorder the terms:
-13.5 + t = -13.793114224

Solving
-13.5 + t = -13.793114224

Solving for variable 't'.

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

Add '13.5' to each side of the equation.
-13.5 + 13.5 + t = -13.793114224 + 13.5

Combine like terms: -13.5 + 13.5 = 0.0
0.0 + t = -13.793114224 + 13.5
t = -13.793114224 + 13.5

Combine like terms: -13.793114224 + 13.5 = -0.293114224
t = -0.293114224

Simplifying
t = -0.293114224


Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {27.293114224, -0.293114224}
Solution
t = {27.293114224, -0.293114224}

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