0=4(4t^2-t-120)

Solution for 0=4(4t^2-t-120) equation:

Simplifying
0 = 4(4t2 + -1t + -120)

Reorder the terms:
0 = 4(-120 + -1t + 4t2)
0 = (-120 * 4 + -1t * 4 + 4t2 * 4)
0 = (-480 + -4t + 16t2)

Solving
0 = -480 + -4t + 16t2

Solving for variable 't'.

Combine like terms: 0 + 480 = 480
480 + 4t + -16t2 = -480 + -4t + 16t2 + 480 + 4t + -16t2

Reorder the terms:
480 + 4t + -16t2 = -480 + 480 + -4t + 4t + 16t2 + -16t2

Combine like terms: -480 + 480 = 0
480 + 4t + -16t2 = 0 + -4t + 4t + 16t2 + -16t2
480 + 4t + -16t2 = -4t + 4t + 16t2 + -16t2

Combine like terms: -4t + 4t = 0
480 + 4t + -16t2 = 0 + 16t2 + -16t2
480 + 4t + -16t2 = 16t2 + -16t2

Combine like terms: 16t2 + -16t2 = 0
480 + 4t + -16t2 = 0

Factor out the Greatest Common Factor (GCF), '4'.
4(120 + t + -4t2) = 0

Ignore the factor 4.
Subproblem 1
Set the factor '(120 + t + -4t2)' equal to zero and attempt to solve:

Simplifying
120 + t + -4t2 = 0

Solving
120 + t + -4t2 = 0

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

Divide each side by '-4'.
-30 + -0.25t + t2 = 0

Move the constant term to the right:

Add '30' to each side of the equation.
-30 + -0.25t + 30 + t2 = 0 + 30

Reorder the terms:
-30 + 30 + -0.25t + t2 = 0 + 30

Combine like terms: -30 + 30 = 0
0 + -0.25t + t2 = 0 + 30
-0.25t + t2 = 0 + 30

Combine like terms: 0 + 30 = 30
-0.25t + t2 = 30

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 = 30 + 0.25

Reorder the terms:
0.25 + -0.25t + t2 = 30 + 0.25

Combine like terms: 30 + 0.25 = 30.25
0.25 + -0.25t + t2 = 30.25

Factor a perfect square on the left side:
(t + 0.5)(t + 0.5) = 30.25

Calculate the square root of the right side: 5.5

Break this problem into two subproblems by setting 
(t + 0.5) equal to 5.5 and -5.5.

Subproblem 1
t + 0.5 = 5.5

Simplifying
t + 0.5 = 5.5

Reorder the terms:
0.5 + t = 5.5

Solving
0.5 + t = 5.5

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 = 5.5 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + t = 5.5 + -0.5
t = 5.5 + -0.5

Combine like terms: 5.5 + -0.5 = 5
t = 5

Simplifying
t = 5

Subproblem 2
t + 0.5 = -5.5

Simplifying
t + 0.5 = -5.5

Reorder the terms:
0.5 + t = -5.5

Solving
0.5 + t = -5.5

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 = -5.5 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + t = -5.5 + -0.5
t = -5.5 + -0.5

Combine like terms: -5.5 + -0.5 = -6
t = -6

Simplifying
t = -6

Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {5, -6}
Solution
t = {5, -6}