0=-16t^2+31t+6

Solution for 0=-16t^2+31t+6 equation:

Simplifying
0 = -16t2 + 31t + 6

Reorder the terms:
0 = 6 + 31t + -16t2

Solving
0 = 6 + 31t + -16t2

Solving for variable 't'.

Combine like terms: 0 + -6 = -6
-6 + -31t + 16t2 = 6 + 31t + -16t2 + -6 + -31t + 16t2

Reorder the terms:
-6 + -31t + 16t2 = 6 + -6 + 31t + -31t + -16t2 + 16t2

Combine like terms: 6 + -6 = 0
-6 + -31t + 16t2 = 0 + 31t + -31t + -16t2 + 16t2
-6 + -31t + 16t2 = 31t + -31t + -16t2 + 16t2

Combine like terms: 31t + -31t = 0
-6 + -31t + 16t2 = 0 + -16t2 + 16t2
-6 + -31t + 16t2 = -16t2 + 16t2

Combine like terms: -16t2 + 16t2 = 0
-6 + -31t + 16t2 = 0

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

Divide each side by '16'.
-0.375 + -1.9375t + t2 = 0

Move the constant term to the right:

Add '0.375' to each side of the equation.
-0.375 + -1.9375t + 0.375 + t2 = 0 + 0.375

Reorder the terms:
-0.375 + 0.375 + -1.9375t + t2 = 0 + 0.375

Combine like terms: -0.375 + 0.375 = 0.000
0.000 + -1.9375t + t2 = 0 + 0.375
-1.9375t + t2 = 0 + 0.375

Combine like terms: 0 + 0.375 = 0.375
-1.9375t + t2 = 0.375

The t term is -1.9375t.  Take half its coefficient (-0.96875).
Square it (0.9384765625) and add it to both sides.

Add '0.9384765625' to each side of the equation.
-1.9375t + 0.9384765625 + t2 = 0.375 + 0.9384765625

Reorder the terms:
0.9384765625 + -1.9375t + t2 = 0.375 + 0.9384765625

Combine like terms: 0.375 + 0.9384765625 = 1.3134765625
0.9384765625 + -1.9375t + t2 = 1.3134765625

Factor a perfect square on the left side:
(t + -0.96875)(t + -0.96875) = 1.3134765625

Calculate the square root of the right side: 1.146070051

Break this problem into two subproblems by setting 
(t + -0.96875) equal to 1.146070051 and -1.146070051.

Subproblem 1
t + -0.96875 = 1.146070051

Simplifying
t + -0.96875 = 1.146070051

Reorder the terms:
-0.96875 + t = 1.146070051

Solving
-0.96875 + t = 1.146070051

Solving for variable 't'.

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

Add '0.96875' to each side of the equation.
-0.96875 + 0.96875 + t = 1.146070051 + 0.96875

Combine like terms: -0.96875 + 0.96875 = 0.00000
0.00000 + t = 1.146070051 + 0.96875
t = 1.146070051 + 0.96875

Combine like terms: 1.146070051 + 0.96875 = 2.114820051
t = 2.114820051

Simplifying
t = 2.114820051

Subproblem 2
t + -0.96875 = -1.146070051

Simplifying
t + -0.96875 = -1.146070051

Reorder the terms:
-0.96875 + t = -1.146070051

Solving
-0.96875 + t = -1.146070051

Solving for variable 't'.

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

Add '0.96875' to each side of the equation.
-0.96875 + 0.96875 + t = -1.146070051 + 0.96875

Combine like terms: -0.96875 + 0.96875 = 0.00000
0.00000 + t = -1.146070051 + 0.96875
t = -1.146070051 + 0.96875

Combine like terms: -1.146070051 + 0.96875 = -0.177320051
t = -0.177320051

Simplifying
t = -0.177320051

Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {2.114820051, -0.177320051}