=-16t^2+32t+19

Solution for =-16t^2+32t+19 equation:

Simplifying
0 = -16t2 + 32t + 19

Reorder the terms:
0 = 19 + 32t + -16t2

Solving
0 = 19 + 32t + -16t2

Solving for variable 't'.

Combine like terms: 0 + -19 = -19
-19 + -32t + 16t2 = 19 + 32t + -16t2 + -19 + -32t + 16t2

Reorder the terms:
-19 + -32t + 16t2 = 19 + -19 + 32t + -32t + -16t2 + 16t2

Combine like terms: 19 + -19 = 0
-19 + -32t + 16t2 = 0 + 32t + -32t + -16t2 + 16t2
-19 + -32t + 16t2 = 32t + -32t + -16t2 + 16t2

Combine like terms: 32t + -32t = 0
-19 + -32t + 16t2 = 0 + -16t2 + 16t2
-19 + -32t + 16t2 = -16t2 + 16t2

Combine like terms: -16t2 + 16t2 = 0
-19 + -32t + 16t2 = 0

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

Divide each side by '16'.
-1.1875 + -2t + t2 = 0

Move the constant term to the right:

Add '1.1875' to each side of the equation.
-1.1875 + -2t + 1.1875 + t2 = 0 + 1.1875

Reorder the terms:
-1.1875 + 1.1875 + -2t + t2 = 0 + 1.1875

Combine like terms: -1.1875 + 1.1875 = 0.0000
0.0000 + -2t + t2 = 0 + 1.1875
-2t + t2 = 0 + 1.1875

Combine like terms: 0 + 1.1875 = 1.1875
-2t + t2 = 1.1875

The t term is -2t.  Take half its coefficient (-1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
-2t + 1 + t2 = 1.1875 + 1

Reorder the terms:
1 + -2t + t2 = 1.1875 + 1

Combine like terms: 1.1875 + 1 = 2.1875
1 + -2t + t2 = 2.1875

Factor a perfect square on the left side:
(t + -1)(t + -1) = 2.1875

Calculate the square root of the right side: 1.479019946

Break this problem into two subproblems by setting 
(t + -1) equal to 1.479019946 and -1.479019946.

Subproblem 1
t + -1 = 1.479019946

Simplifying
t + -1 = 1.479019946

Reorder the terms:
-1 + t = 1.479019946

Solving
-1 + t = 1.479019946

Solving for variable 't'.

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

Add '1' to each side of the equation.
-1 + 1 + t = 1.479019946 + 1

Combine like terms: -1 + 1 = 0
0 + t = 1.479019946 + 1
t = 1.479019946 + 1

Combine like terms: 1.479019946 + 1 = 2.479019946
t = 2.479019946

Simplifying
t = 2.479019946

Subproblem 2
t + -1 = -1.479019946

Simplifying
t + -1 = -1.479019946

Reorder the terms:
-1 + t = -1.479019946

Solving
-1 + t = -1.479019946

Solving for variable 't'.

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

Add '1' to each side of the equation.
-1 + 1 + t = -1.479019946 + 1

Combine like terms: -1 + 1 = 0
0 + t = -1.479019946 + 1
t = -1.479019946 + 1

Combine like terms: -1.479019946 + 1 = -0.479019946
t = -0.479019946

Simplifying
t = -0.479019946

Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {2.479019946, -0.479019946}