32=48t+16t^2

Solution for 32=48t+16t^2 equation:

Simplifying
32 = 48t + 16t2

Solving
32 = 48t + 16t2

Solving for variable 't'.

Reorder the terms:
32 + -48t + -16t2 = 48t + -48t + 16t2 + -16t2

Combine like terms: 48t + -48t = 0
32 + -48t + -16t2 = 0 + 16t2 + -16t2
32 + -48t + -16t2 = 16t2 + -16t2

Combine like terms: 16t2 + -16t2 = 0
32 + -48t + -16t2 = 0

Factor out the Greatest Common Factor (GCF), '16'.
16(2 + -3t + -1t2) = 0

Ignore the factor 16.
Subproblem 1
Set the factor '(2 + -3t + -1t2)' equal to zero and attempt to solve:

Simplifying
2 + -3t + -1t2 = 0

Solving
2 + -3t + -1t2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-2 + 3t + t2 = 0

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + 3t + 2 + t2 = 0 + 2

Reorder the terms:
-2 + 2 + 3t + t2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + 3t + t2 = 0 + 2
3t + t2 = 0 + 2

Combine like terms: 0 + 2 = 2
3t + t2 = 2

The t term is 3t.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3t + 2.25 + t2 = 2 + 2.25

Reorder the terms:
2.25 + 3t + t2 = 2 + 2.25

Combine like terms: 2 + 2.25 = 4.25
2.25 + 3t + t2 = 4.25

Factor a perfect square on the left side:
(t + 1.5)(t + 1.5) = 4.25

Calculate the square root of the right side: 2.061552813

Break this problem into two subproblems by setting 
(t + 1.5) equal to 2.061552813 and -2.061552813.

Subproblem 1
t + 1.5 = 2.061552813

Simplifying
t + 1.5 = 2.061552813

Reorder the terms:
1.5 + t = 2.061552813

Solving
1.5 + t = 2.061552813

Solving for variable 't'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + t = 2.061552813 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + t = 2.061552813 + -1.5
t = 2.061552813 + -1.5

Combine like terms: 2.061552813 + -1.5 = 0.561552813
t = 0.561552813

Simplifying
t = 0.561552813

Subproblem 2
t + 1.5 = -2.061552813

Simplifying
t + 1.5 = -2.061552813

Reorder the terms:
1.5 + t = -2.061552813

Solving
1.5 + t = -2.061552813

Solving for variable 't'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + t = -2.061552813 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + t = -2.061552813 + -1.5
t = -2.061552813 + -1.5

Combine like terms: -2.061552813 + -1.5 = -3.561552813
t = -3.561552813

Simplifying
t = -3.561552813

Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {0.561552813, -3.561552813}
Solution
t = {0.561552813, -3.561552813}