16t^2-202t+82=0

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Solution for 16t^2-202t+82=0 equation: 


Simplifying
16t2 + -202t + 82 = 0

Reorder the terms:
82 + -202t + 16t2 = 0

Solving
82 + -202t + 16t2 = 0

Solving for variable 't'.

Factor out the Greatest Common Factor (GCF), '2'.
2(41 + -101t + 8t2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(41 + -101t + 8t2)' equal to zero and attempt to solve:

Simplifying
41 + -101t + 8t2 = 0

Solving
41 + -101t + 8t2 = 0

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

Divide each side by '8'.
5.125 + -12.625t + t2 = 0

Move the constant term to the right:

Add '-5.125' to each side of the equation.
5.125 + -12.625t + -5.125 + t2 = 0 + -5.125

Reorder the terms:
5.125 + -5.125 + -12.625t + t2 = 0 + -5.125

Combine like terms: 5.125 + -5.125 = 0.000
0.000 + -12.625t + t2 = 0 + -5.125
-12.625t + t2 = 0 + -5.125

Combine like terms: 0 + -5.125 = -5.125
-12.625t + t2 = -5.125

The t term is -12.625t.  Take half its coefficient (-6.3125).
Square it (39.84765625) and add it to both sides.

Add '39.84765625' to each side of the equation.
-12.625t + 39.84765625 + t2 = -5.125 + 39.84765625

Reorder the terms:
39.84765625 + -12.625t + t2 = -5.125 + 39.84765625

Combine like terms: -5.125 + 39.84765625 = 34.72265625
39.84765625 + -12.625t + t2 = 34.72265625

Factor a perfect square on the left side:
(t + -6.3125)(t + -6.3125) = 34.72265625

Calculate the square root of the right side: 5.892593338

Break this problem into two subproblems by setting 
(t + -6.3125) equal to 5.892593338 and -5.892593338.


Subproblem 1
t + -6.3125 = 5.892593338

Simplifying
t + -6.3125 = 5.892593338

Reorder the terms:
-6.3125 + t = 5.892593338

Solving
-6.3125 + t = 5.892593338

Solving for variable 't'.

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

Add '6.3125' to each side of the equation.
-6.3125 + 6.3125 + t = 5.892593338 + 6.3125

Combine like terms: -6.3125 + 6.3125 = 0.0000
0.0000 + t = 5.892593338 + 6.3125
t = 5.892593338 + 6.3125

Combine like terms: 5.892593338 + 6.3125 = 12.205093338
t = 12.205093338

Simplifying
t = 12.205093338


Subproblem 2
t + -6.3125 = -5.892593338

Simplifying
t + -6.3125 = -5.892593338

Reorder the terms:
-6.3125 + t = -5.892593338

Solving
-6.3125 + t = -5.892593338

Solving for variable 't'.

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

Add '6.3125' to each side of the equation.
-6.3125 + 6.3125 + t = -5.892593338 + 6.3125

Combine like terms: -6.3125 + 6.3125 = 0.0000
0.0000 + t = -5.892593338 + 6.3125
t = -5.892593338 + 6.3125

Combine like terms: -5.892593338 + 6.3125 = 0.419906662
t = 0.419906662

Simplifying
t = 0.419906662


Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {12.205093338, 0.419906662}
Solution
t = {12.205093338, 0.419906662}

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