16t^2+4t-500=0

Solution for 16t^2+4t-500=0 equation:

Simplifying
16t2 + 4t + -500 = 0

Reorder the terms:
-500 + 4t + 16t2 = 0

Solving
-500 + 4t + 16t2 = 0

Solving for variable 't'.

Factor out the Greatest Common Factor (GCF), '4'.
4(-125 + t + 4t2) = 0

Ignore the factor 4.
Subproblem 1
Set the factor '(-125 + t + 4t2)' equal to zero and attempt to solve:

Simplifying
-125 + t + 4t2 = 0

Solving
-125 + t + 4t2 = 0

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

Divide each side by '4'.
-31.25 + 0.25t + t2 = 0

Move the constant term to the right:

Add '31.25' to each side of the equation.
-31.25 + 0.25t + 31.25 + t2 = 0 + 31.25

Reorder the terms:
-31.25 + 31.25 + 0.25t + t2 = 0 + 31.25

Combine like terms: -31.25 + 31.25 = 0.00
0.00 + 0.25t + t2 = 0 + 31.25
0.25t + t2 = 0 + 31.25

Combine like terms: 0 + 31.25 = 31.25
0.25t + t2 = 31.25

The t term is t.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
0.25t + 0.25 + t2 = 31.25 + 0.25

Reorder the terms:
0.25 + 0.25t + t2 = 31.25 + 0.25

Combine like terms: 31.25 + 0.25 = 31.5
0.25 + 0.25t + t2 = 31.5

Factor a perfect square on the left side:
(t + 0.5)(t + 0.5) = 31.5

Calculate the square root of the right side: 5.61248608

Break this problem into two subproblems by setting 
(t + 0.5) equal to 5.61248608 and -5.61248608.

Subproblem 1
t + 0.5 = 5.61248608

Simplifying
t + 0.5 = 5.61248608

Reorder the terms:
0.5 + t = 5.61248608

Solving
0.5 + t = 5.61248608

Solving for variable 't'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + t = 5.61248608 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + t = 5.61248608 + -0.5
t = 5.61248608 + -0.5

Combine like terms: 5.61248608 + -0.5 = 5.11248608
t = 5.11248608

Simplifying
t = 5.11248608

Subproblem 2
t + 0.5 = -5.61248608

Simplifying
t + 0.5 = -5.61248608

Reorder the terms:
0.5 + t = -5.61248608

Solving
0.5 + t = -5.61248608

Solving for variable 't'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + t = -5.61248608 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + t = -5.61248608 + -0.5
t = -5.61248608 + -0.5

Combine like terms: -5.61248608 + -0.5 = -6.11248608
t = -6.11248608

Simplifying
t = -6.11248608

Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {5.11248608, -6.11248608}
Solution
t = {5.11248608, -6.11248608}