16t^2+122t+10=0

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Solution for 16t^2+122t+10=0 equation: 


Simplifying
16t2 + 122t + 10 = 0

Reorder the terms:
10 + 122t + 16t2 = 0

Solving
10 + 122t + 16t2 = 0

Solving for variable 't'.

Factor out the Greatest Common Factor (GCF), '2'.
2(5 + 61t + 8t2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(5 + 61t + 8t2)' equal to zero and attempt to solve:

Simplifying
5 + 61t + 8t2 = 0

Solving
5 + 61t + 8t2 = 0

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

Divide each side by '8'.
0.625 + 7.625t + t2 = 0

Move the constant term to the right:

Add '-0.625' to each side of the equation.
0.625 + 7.625t + -0.625 + t2 = 0 + -0.625

Reorder the terms:
0.625 + -0.625 + 7.625t + t2 = 0 + -0.625

Combine like terms: 0.625 + -0.625 = 0.000
0.000 + 7.625t + t2 = 0 + -0.625
7.625t + t2 = 0 + -0.625

Combine like terms: 0 + -0.625 = -0.625
7.625t + t2 = -0.625

The t term is 7.625t.  Take half its coefficient (3.8125).
Square it (14.53515625) and add it to both sides.

Add '14.53515625' to each side of the equation.
7.625t + 14.53515625 + t2 = -0.625 + 14.53515625

Reorder the terms:
14.53515625 + 7.625t + t2 = -0.625 + 14.53515625

Combine like terms: -0.625 + 14.53515625 = 13.91015625
14.53515625 + 7.625t + t2 = 13.91015625

Factor a perfect square on the left side:
(t + 3.8125)(t + 3.8125) = 13.91015625

Calculate the square root of the right side: 3.729632187

Break this problem into two subproblems by setting 
(t + 3.8125) equal to 3.729632187 and -3.729632187.


Subproblem 1
t + 3.8125 = 3.729632187

Simplifying
t + 3.8125 = 3.729632187

Reorder the terms:
3.8125 + t = 3.729632187

Solving
3.8125 + t = 3.729632187

Solving for variable 't'.

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

Add '-3.8125' to each side of the equation.
3.8125 + -3.8125 + t = 3.729632187 + -3.8125

Combine like terms: 3.8125 + -3.8125 = 0.0000
0.0000 + t = 3.729632187 + -3.8125
t = 3.729632187 + -3.8125

Combine like terms: 3.729632187 + -3.8125 = -0.082867813
t = -0.082867813

Simplifying
t = -0.082867813


Subproblem 2
t + 3.8125 = -3.729632187

Simplifying
t + 3.8125 = -3.729632187

Reorder the terms:
3.8125 + t = -3.729632187

Solving
3.8125 + t = -3.729632187

Solving for variable 't'.

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

Add '-3.8125' to each side of the equation.
3.8125 + -3.8125 + t = -3.729632187 + -3.8125

Combine like terms: 3.8125 + -3.8125 = 0.0000
0.0000 + t = -3.729632187 + -3.8125
t = -3.729632187 + -3.8125

Combine like terms: -3.729632187 + -3.8125 = -7.542132187
t = -7.542132187

Simplifying
t = -7.542132187


Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {-0.082867813, -7.542132187}
Solution
t = {-0.082867813, -7.542132187}

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