9.8t^2-32t+20.5=0

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Solution for 9.8t^2-32t+20.5=0 equation: 


Simplifying
9.8t2 + -32t + 20.5 = 0

Reorder the terms:
20.5 + -32t + 9.8t2 = 0

Solving
20.5 + -32t + 9.8t2 = 0

Solving for variable 't'.

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

Divide each side by '9.8'.
2.091836735 + -3.265306122t + t2 = 0

Move the constant term to the right:

Add '-2.091836735' to each side of the equation.
2.091836735 + -3.265306122t + -2.091836735 + t2 = 0 + -2.091836735

Reorder the terms:
2.091836735 + -2.091836735 + -3.265306122t + t2 = 0 + -2.091836735

Combine like terms: 2.091836735 + -2.091836735 = 0.000000000
0.000000000 + -3.265306122t + t2 = 0 + -2.091836735
-3.265306122t + t2 = 0 + -2.091836735

Combine like terms: 0 + -2.091836735 = -2.091836735
-3.265306122t + t2 = -2.091836735

The t term is -3.265306122t.  Take half its coefficient (-1.632653061).
Square it (2.665556018) and add it to both sides.

Add '2.665556018' to each side of the equation.
-3.265306122t + 2.665556018 + t2 = -2.091836735 + 2.665556018

Reorder the terms:
2.665556018 + -3.265306122t + t2 = -2.091836735 + 2.665556018

Combine like terms: -2.091836735 + 2.665556018 = 0.573719283
2.665556018 + -3.265306122t + t2 = 0.573719283

Factor a perfect square on the left side:
(t + -1.632653061)(t + -1.632653061) = 0.573719283

Calculate the square root of the right side: 0.757442594

Break this problem into two subproblems by setting 
(t + -1.632653061) equal to 0.757442594 and -0.757442594.


Subproblem 1
t + -1.632653061 = 0.757442594

Simplifying
t + -1.632653061 = 0.757442594

Reorder the terms:
-1.632653061 + t = 0.757442594

Solving
-1.632653061 + t = 0.757442594

Solving for variable 't'.

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

Add '1.632653061' to each side of the equation.
-1.632653061 + 1.632653061 + t = 0.757442594 + 1.632653061

Combine like terms: -1.632653061 + 1.632653061 = 0.000000000
0.000000000 + t = 0.757442594 + 1.632653061
t = 0.757442594 + 1.632653061

Combine like terms: 0.757442594 + 1.632653061 = 2.390095655
t = 2.390095655

Simplifying
t = 2.390095655


Subproblem 2
t + -1.632653061 = -0.757442594

Simplifying
t + -1.632653061 = -0.757442594

Reorder the terms:
-1.632653061 + t = -0.757442594

Solving
-1.632653061 + t = -0.757442594

Solving for variable 't'.

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

Add '1.632653061' to each side of the equation.
-1.632653061 + 1.632653061 + t = -0.757442594 + 1.632653061

Combine like terms: -1.632653061 + 1.632653061 = 0.000000000
0.000000000 + t = -0.757442594 + 1.632653061
t = -0.757442594 + 1.632653061

Combine like terms: -0.757442594 + 1.632653061 = 0.875210467
t = 0.875210467

Simplifying
t = 0.875210467


Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {2.390095655, 0.875210467}

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