20.3136t^2-108.378t+125.12=0

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Solution for 20.3136t^2-108.378t+125.12=0 equation: 


Simplifying
20.3136t2 + -108.378t + 125.12 = 0

Reorder the terms:
125.12 + -108.378t + 20.3136t2 = 0

Solving
125.12 + -108.378t + 20.3136t2 = 0

Solving for variable 't'.

Begin completing the square.  Divide all terms by
20.3136 the coefficient of the squared term: 

Divide each side by '20.3136'.
6.15942029 + -5.335243384t + t2 = 0

Move the constant term to the right:

Add '-6.15942029' to each side of the equation.
6.15942029 + -5.335243384t + -6.15942029 + t2 = 0 + -6.15942029

Reorder the terms:
6.15942029 + -6.15942029 + -5.335243384t + t2 = 0 + -6.15942029

Combine like terms: 6.15942029 + -6.15942029 = 0.00000000
0.00000000 + -5.335243384t + t2 = 0 + -6.15942029
-5.335243384t + t2 = 0 + -6.15942029

Combine like terms: 0 + -6.15942029 = -6.15942029
-5.335243384t + t2 = -6.15942029

The t term is -5.335243384t.  Take half its coefficient (-2.667621692).
Square it (7.116205492) and add it to both sides.

Add '7.116205492' to each side of the equation.
-5.335243384t + 7.116205492 + t2 = -6.15942029 + 7.116205492

Reorder the terms:
7.116205492 + -5.335243384t + t2 = -6.15942029 + 7.116205492

Combine like terms: -6.15942029 + 7.116205492 = 0.956785202
7.116205492 + -5.335243384t + t2 = 0.956785202

Factor a perfect square on the left side:
(t + -2.667621692)(t + -2.667621692) = 0.956785202

Calculate the square root of the right side: 0.978153977

Break this problem into two subproblems by setting 
(t + -2.667621692) equal to 0.978153977 and -0.978153977.


Subproblem 1
t + -2.667621692 = 0.978153977

Simplifying
t + -2.667621692 = 0.978153977

Reorder the terms:
-2.667621692 + t = 0.978153977

Solving
-2.667621692 + t = 0.978153977

Solving for variable 't'.

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

Add '2.667621692' to each side of the equation.
-2.667621692 + 2.667621692 + t = 0.978153977 + 2.667621692

Combine like terms: -2.667621692 + 2.667621692 = 0.000000000
0.000000000 + t = 0.978153977 + 2.667621692
t = 0.978153977 + 2.667621692

Combine like terms: 0.978153977 + 2.667621692 = 3.645775669
t = 3.645775669

Simplifying
t = 3.645775669


Subproblem 2
t + -2.667621692 = -0.978153977

Simplifying
t + -2.667621692 = -0.978153977

Reorder the terms:
-2.667621692 + t = -0.978153977

Solving
-2.667621692 + t = -0.978153977

Solving for variable 't'.

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

Add '2.667621692' to each side of the equation.
-2.667621692 + 2.667621692 + t = -0.978153977 + 2.667621692

Combine like terms: -2.667621692 + 2.667621692 = 0.000000000
0.000000000 + t = -0.978153977 + 2.667621692
t = -0.978153977 + 2.667621692

Combine like terms: -0.978153977 + 2.667621692 = 1.689467715
t = 1.689467715

Simplifying
t = 1.689467715


Solution
The solution to the problem is based on the solutions
from the subproblems.
t = {3.645775669, 1.689467715}

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