108=8+96t-16(t^2)

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Solution for 108=8+96t-16(t^2) equation: 


Simplifying
108 = 8 + 96t + -16(t2)

Solving
108 = 8 + 96t + -16t2

Solving for variable 't'.

Combine like terms: 108 + -8 = 100
100 + -96t + 16t2 = 8 + 96t + -16t2 + -8 + -96t + 16t2

Reorder the terms:
100 + -96t + 16t2 = 8 + -8 + 96t + -96t + -16t2 + 16t2

Combine like terms: 8 + -8 = 0
100 + -96t + 16t2 = 0 + 96t + -96t + -16t2 + 16t2
100 + -96t + 16t2 = 96t + -96t + -16t2 + 16t2

Combine like terms: 96t + -96t = 0
100 + -96t + 16t2 = 0 + -16t2 + 16t2
100 + -96t + 16t2 = -16t2 + 16t2

Combine like terms: -16t2 + 16t2 = 0
100 + -96t + 16t2 = 0

Factor out the Greatest Common Factor (GCF), '4'.
4(25 + -24t + 4t2) = 0

Ignore the factor 4.

Subproblem 1
Set the factor '(25 + -24t + 4t2)' equal to zero and attempt to solve:

Simplifying
25 + -24t + 4t2 = 0

Solving
25 + -24t + 4t2 = 0

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

Divide each side by '4'.
6.25 + -6t + t2 = 0

Move the constant term to the right:

Add '-6.25' to each side of the equation.
6.25 + -6t + -6.25 + t2 = 0 + -6.25

Reorder the terms:
6.25 + -6.25 + -6t + t2 = 0 + -6.25

Combine like terms: 6.25 + -6.25 = 0.00
0.00 + -6t + t2 = 0 + -6.25
-6t + t2 = 0 + -6.25

Combine like terms: 0 + -6.25 = -6.25
-6t + t2 = -6.25

The t term is -6t.  Take half its coefficient (-3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
-6t + 9 + t2 = -6.25 + 9

Reorder the terms:
9 + -6t + t2 = -6.25 + 9

Combine like terms: -6.25 + 9 = 2.75
9 + -6t + t2 = 2.75

Factor a perfect square on the left side:
(t + -3)(t + -3) = 2.75

Calculate the square root of the right side: 1.658312395

Break this problem into two subproblems by setting 
(t + -3) equal to 1.658312395 and -1.658312395.


Subproblem 1
t + -3 = 1.658312395

Simplifying
t + -3 = 1.658312395

Reorder the terms:
-3 + t = 1.658312395

Solving
-3 + t = 1.658312395

Solving for variable 't'.

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

Add '3' to each side of the equation.
-3 + 3 + t = 1.658312395 + 3

Combine like terms: -3 + 3 = 0
0 + t = 1.658312395 + 3
t = 1.658312395 + 3

Combine like terms: 1.658312395 + 3 = 4.658312395
t = 4.658312395

Simplifying
t = 4.658312395


Subproblem 2
t + -3 = -1.658312395

Simplifying
t + -3 = -1.658312395

Reorder the terms:
-3 + t = -1.658312395

Solving
-3 + t = -1.658312395

Solving for variable 't'.

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

Add '3' to each side of the equation.
-3 + 3 + t = -1.658312395 + 3

Combine like terms: -3 + 3 = 0
0 + t = -1.658312395 + 3
t = -1.658312395 + 3

Combine like terms: -1.658312395 + 3 = 1.341687605
t = 1.341687605

Simplifying
t = 1.341687605


Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {4.658312395, 1.341687605}
Solution
t = {4.658312395, 1.341687605}

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