20(t)=20-16t^2+32

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Solution for 20(t)=20-16t^2+32 equation: 


Simplifying
20(t) = 20 + -16t2 + 32

Reorder the terms:
20t = 20 + 32 + -16t2

Combine like terms: 20 + 32 = 52
20t = 52 + -16t2

Solving
20t = 52 + -16t2

Solving for variable 't'.

Reorder the terms:
-52 + 20t + 16t2 = 52 + -16t2 + -52 + 16t2

Reorder the terms:
-52 + 20t + 16t2 = 52 + -52 + -16t2 + 16t2

Combine like terms: 52 + -52 = 0
-52 + 20t + 16t2 = 0 + -16t2 + 16t2
-52 + 20t + 16t2 = -16t2 + 16t2

Combine like terms: -16t2 + 16t2 = 0
-52 + 20t + 16t2 = 0

Factor out the Greatest Common Factor (GCF), '4'.
4(-13 + 5t + 4t2) = 0

Ignore the factor 4.

Subproblem 1
Set the factor '(-13 + 5t + 4t2)' equal to zero and attempt to solve:

Simplifying
-13 + 5t + 4t2 = 0

Solving
-13 + 5t + 4t2 = 0

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

Divide each side by '4'.
-3.25 + 1.25t + t2 = 0

Move the constant term to the right:

Add '3.25' to each side of the equation.
-3.25 + 1.25t + 3.25 + t2 = 0 + 3.25

Reorder the terms:
-3.25 + 3.25 + 1.25t + t2 = 0 + 3.25

Combine like terms: -3.25 + 3.25 = 0.00
0.00 + 1.25t + t2 = 0 + 3.25
1.25t + t2 = 0 + 3.25

Combine like terms: 0 + 3.25 = 3.25
1.25t + t2 = 3.25

The t term is 1.25t.  Take half its coefficient (0.625).
Square it (0.390625) and add it to both sides.

Add '0.390625' to each side of the equation.
1.25t + 0.390625 + t2 = 3.25 + 0.390625

Reorder the terms:
0.390625 + 1.25t + t2 = 3.25 + 0.390625

Combine like terms: 3.25 + 0.390625 = 3.640625
0.390625 + 1.25t + t2 = 3.640625

Factor a perfect square on the left side:
(t + 0.625)(t + 0.625) = 3.640625

Calculate the square root of the right side: 1.90804219

Break this problem into two subproblems by setting 
(t + 0.625) equal to 1.90804219 and -1.90804219.


Subproblem 1
t + 0.625 = 1.90804219

Simplifying
t + 0.625 = 1.90804219

Reorder the terms:
0.625 + t = 1.90804219

Solving
0.625 + t = 1.90804219

Solving for variable 't'.

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

Add '-0.625' to each side of the equation.
0.625 + -0.625 + t = 1.90804219 + -0.625

Combine like terms: 0.625 + -0.625 = 0.000
0.000 + t = 1.90804219 + -0.625
t = 1.90804219 + -0.625

Combine like terms: 1.90804219 + -0.625 = 1.28304219
t = 1.28304219

Simplifying
t = 1.28304219


Subproblem 2
t + 0.625 = -1.90804219

Simplifying
t + 0.625 = -1.90804219

Reorder the terms:
0.625 + t = -1.90804219

Solving
0.625 + t = -1.90804219

Solving for variable 't'.

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

Add '-0.625' to each side of the equation.
0.625 + -0.625 + t = -1.90804219 + -0.625

Combine like terms: 0.625 + -0.625 = 0.000
0.000 + t = -1.90804219 + -0.625
t = -1.90804219 + -0.625

Combine like terms: -1.90804219 + -0.625 = -2.53304219
t = -2.53304219

Simplifying
t = -2.53304219


Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {1.28304219, -2.53304219}
Solution
t = {1.28304219, -2.53304219}

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