-.5t^2-3t-4=0

Solution for -.5t^2-3t-4=0 equation:

Simplifying
-0.5t2 + -3t + -4 = 0

Reorder the terms:
-4 + -3t + -0.5t2 = 0

Solving
-4 + -3t + -0.5t2 = 0

Solving for variable 't'.

Begin completing the square.  Divide all terms by
-0.5 the coefficient of the squared term: 

Divide each side by '-0.5'.
8 + 6t + t2 = 0

Move the constant term to the right:

Add '-8' to each side of the equation.
8 + 6t + -8 + t2 = 0 + -8

Reorder the terms:
8 + -8 + 6t + t2 = 0 + -8

Combine like terms: 8 + -8 = 0
0 + 6t + t2 = 0 + -8
6t + t2 = 0 + -8

Combine like terms: 0 + -8 = -8
6t + t2 = -8

The t term is 6t.  Take half its coefficient (3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
6t + 9 + t2 = -8 + 9

Reorder the terms:
9 + 6t + t2 = -8 + 9

Combine like terms: -8 + 9 = 1
9 + 6t + t2 = 1

Factor a perfect square on the left side:
(t + 3)(t + 3) = 1

Calculate the square root of the right side: 1

Break this problem into two subproblems by setting 
(t + 3) equal to 1 and -1.

Subproblem 1
t + 3 = 1

Simplifying
t + 3 = 1

Reorder the terms:
3 + t = 1

Solving
3 + t = 1

Solving for variable 't'.

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

Add '-3' to each side of the equation.
3 + -3 + t = 1 + -3

Combine like terms: 3 + -3 = 0
0 + t = 1 + -3
t = 1 + -3

Combine like terms: 1 + -3 = -2
t = -2

Simplifying
t = -2

Subproblem 2
t + 3 = -1

Simplifying
t + 3 = -1

Reorder the terms:
3 + t = -1

Solving
3 + t = -1

Solving for variable 't'.

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

Add '-3' to each side of the equation.
3 + -3 + t = -1 + -3

Combine like terms: 3 + -3 = 0
0 + t = -1 + -3
t = -1 + -3

Combine like terms: -1 + -3 = -4
t = -4

Simplifying
t = -4

Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {-2, -4}