t=20-16t^2+32t

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


Simplifying
t = 20 + -16t2 + 32t

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

Solving
t = 20 + 32t + -16t2

Solving for variable 't'.

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

Combine like terms: t + -32t = -31t
-20 + -31t + 16t2 = 20 + 32t + -16t2 + -20 + -32t + 16t2

Reorder the terms:
-20 + -31t + 16t2 = 20 + -20 + 32t + -32t + -16t2 + 16t2

Combine like terms: 20 + -20 = 0
-20 + -31t + 16t2 = 0 + 32t + -32t + -16t2 + 16t2
-20 + -31t + 16t2 = 32t + -32t + -16t2 + 16t2

Combine like terms: 32t + -32t = 0
-20 + -31t + 16t2 = 0 + -16t2 + 16t2
-20 + -31t + 16t2 = -16t2 + 16t2

Combine like terms: -16t2 + 16t2 = 0
-20 + -31t + 16t2 = 0

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

Divide each side by '16'.
-1.25 + -1.9375t + t2 = 0

Move the constant term to the right:

Add '1.25' to each side of the equation.
-1.25 + -1.9375t + 1.25 + t2 = 0 + 1.25

Reorder the terms:
-1.25 + 1.25 + -1.9375t + t2 = 0 + 1.25

Combine like terms: -1.25 + 1.25 = 0.00
0.00 + -1.9375t + t2 = 0 + 1.25
-1.9375t + t2 = 0 + 1.25

Combine like terms: 0 + 1.25 = 1.25
-1.9375t + t2 = 1.25

The t term is -1.9375t.  Take half its coefficient (-0.96875).
Square it (0.9384765625) and add it to both sides.

Add '0.9384765625' to each side of the equation.
-1.9375t + 0.9384765625 + t2 = 1.25 + 0.9384765625

Reorder the terms:
0.9384765625 + -1.9375t + t2 = 1.25 + 0.9384765625

Combine like terms: 1.25 + 0.9384765625 = 2.1884765625
0.9384765625 + -1.9375t + t2 = 2.1884765625

Factor a perfect square on the left side:
(t + -0.96875)(t + -0.96875) = 2.1884765625

Calculate the square root of the right side: 1.479350047

Break this problem into two subproblems by setting 
(t + -0.96875) equal to 1.479350047 and -1.479350047.

Subproblem 1

t + -0.96875 = 1.479350047 Simplifying t + -0.96875 = 1.479350047 Reorder the terms: -0.96875 + t = 1.479350047 Solving -0.96875 + t = 1.479350047 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.96875' to each side of the equation. -0.96875 + 0.96875 + t = 1.479350047 + 0.96875 Combine like terms: -0.96875 + 0.96875 = 0.00000 0.00000 + t = 1.479350047 + 0.96875 t = 1.479350047 + 0.96875 Combine like terms: 1.479350047 + 0.96875 = 2.448100047 t = 2.448100047 Simplifying t = 2.448100047

Subproblem 2

t + -0.96875 = -1.479350047 Simplifying t + -0.96875 = -1.479350047 Reorder the terms: -0.96875 + t = -1.479350047 Solving -0.96875 + t = -1.479350047 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.96875' to each side of the equation. -0.96875 + 0.96875 + t = -1.479350047 + 0.96875 Combine like terms: -0.96875 + 0.96875 = 0.00000 0.00000 + t = -1.479350047 + 0.96875 t = -1.479350047 + 0.96875 Combine like terms: -1.479350047 + 0.96875 = -0.510600047 t = -0.510600047 Simplifying t = -0.510600047

Solution

The solution to the problem is based on the solutions from the subproblems. t = {2.448100047, -0.510600047}

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