400(t)=16t^2+32t+3

Solution for 400(t)=16t^2+32t+3 equation:

Simplifying
400(t) = 16t2 + 32t + 3

Reorder the terms:
400t = 3 + 32t + 16t2

Solving
400t = 3 + 32t + 16t2

Solving for variable 't'.

Reorder the terms:
-3 + 400t + -32t + -16t2 = 3 + 32t + 16t2 + -3 + -32t + -16t2

Combine like terms: 400t + -32t = 368t
-3 + 368t + -16t2 = 3 + 32t + 16t2 + -3 + -32t + -16t2

Reorder the terms:
-3 + 368t + -16t2 = 3 + -3 + 32t + -32t + 16t2 + -16t2

Combine like terms: 3 + -3 = 0
-3 + 368t + -16t2 = 0 + 32t + -32t + 16t2 + -16t2
-3 + 368t + -16t2 = 32t + -32t + 16t2 + -16t2

Combine like terms: 32t + -32t = 0
-3 + 368t + -16t2 = 0 + 16t2 + -16t2
-3 + 368t + -16t2 = 16t2 + -16t2

Combine like terms: 16t2 + -16t2 = 0
-3 + 368t + -16t2 = 0

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

Divide each side by '-16'.
0.1875 + -23t + t2 = 0

Move the constant term to the right:

Add '-0.1875' to each side of the equation.
0.1875 + -23t + -0.1875 + t2 = 0 + -0.1875

Reorder the terms:
0.1875 + -0.1875 + -23t + t2 = 0 + -0.1875

Combine like terms: 0.1875 + -0.1875 = 0.0000
0.0000 + -23t + t2 = 0 + -0.1875
-23t + t2 = 0 + -0.1875

Combine like terms: 0 + -0.1875 = -0.1875
-23t + t2 = -0.1875

The t term is -23t.  Take half its coefficient (-11.5).
Square it (132.25) and add it to both sides.

Add '132.25' to each side of the equation.
-23t + 132.25 + t2 = -0.1875 + 132.25

Reorder the terms:
132.25 + -23t + t2 = -0.1875 + 132.25

Combine like terms: -0.1875 + 132.25 = 132.0625
132.25 + -23t + t2 = 132.0625

Factor a perfect square on the left side:
(t + -11.5)(t + -11.5) = 132.0625

Calculate the square root of the right side: 11.491844935

Break this problem into two subproblems by setting 
(t + -11.5) equal to 11.491844935 and -11.491844935.

Subproblem 1
t + -11.5 = 11.491844935

Simplifying
t + -11.5 = 11.491844935

Reorder the terms:
-11.5 + t = 11.491844935

Solving
-11.5 + t = 11.491844935

Solving for variable 't'.

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

Add '11.5' to each side of the equation.
-11.5 + 11.5 + t = 11.491844935 + 11.5

Combine like terms: -11.5 + 11.5 = 0.0
0.0 + t = 11.491844935 + 11.5
t = 11.491844935 + 11.5

Combine like terms: 11.491844935 + 11.5 = 22.991844935
t = 22.991844935

Simplifying
t = 22.991844935

Subproblem 2
t + -11.5 = -11.491844935

Simplifying
t + -11.5 = -11.491844935

Reorder the terms:
-11.5 + t = -11.491844935

Solving
-11.5 + t = -11.491844935

Solving for variable 't'.

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

Add '11.5' to each side of the equation.
-11.5 + 11.5 + t = -11.491844935 + 11.5

Combine like terms: -11.5 + 11.5 = 0.0
0.0 + t = -11.491844935 + 11.5
t = -11.491844935 + 11.5

Combine like terms: -11.491844935 + 11.5 = 0.008155065
t = 0.008155065

Simplifying
t = 0.008155065

Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {22.991844935, 0.008155065}