0=-16t^2+41t+209

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Solution for 0=-16t^2+41t+209 equation:


Simplifying
0 = -16t2 + 41t + 209

Reorder the terms:
0 = 209 + 41t + -16t2

Solving
0 = 209 + 41t + -16t2

Solving for variable 't'.

Combine like terms: 0 + -209 = -209
-209 + -41t + 16t2 = 209 + 41t + -16t2 + -209 + -41t + 16t2

Reorder the terms:
-209 + -41t + 16t2 = 209 + -209 + 41t + -41t + -16t2 + 16t2

Combine like terms: 209 + -209 = 0
-209 + -41t + 16t2 = 0 + 41t + -41t + -16t2 + 16t2
-209 + -41t + 16t2 = 41t + -41t + -16t2 + 16t2

Combine like terms: 41t + -41t = 0
-209 + -41t + 16t2 = 0 + -16t2 + 16t2
-209 + -41t + 16t2 = -16t2 + 16t2

Combine like terms: -16t2 + 16t2 = 0
-209 + -41t + 16t2 = 0

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

Divide each side by '16'.
-13.0625 + -2.5625t + t2 = 0

Move the constant term to the right:

Add '13.0625' to each side of the equation.
-13.0625 + -2.5625t + 13.0625 + t2 = 0 + 13.0625

Reorder the terms:
-13.0625 + 13.0625 + -2.5625t + t2 = 0 + 13.0625

Combine like terms: -13.0625 + 13.0625 = 0.0000
0.0000 + -2.5625t + t2 = 0 + 13.0625
-2.5625t + t2 = 0 + 13.0625

Combine like terms: 0 + 13.0625 = 13.0625
-2.5625t + t2 = 13.0625

The t term is -2.5625t.  Take half its coefficient (-1.28125).
Square it (1.641601563) and add it to both sides.

Add '1.641601563' to each side of the equation.
-2.5625t + 1.641601563 + t2 = 13.0625 + 1.641601563

Reorder the terms:
1.641601563 + -2.5625t + t2 = 13.0625 + 1.641601563

Combine like terms: 13.0625 + 1.641601563 = 14.704101563
1.641601563 + -2.5625t + t2 = 14.704101563

Factor a perfect square on the left side:
(t + -1.28125)(t + -1.28125) = 14.704101563

Calculate the square root of the right side: 3.834592751

Break this problem into two subproblems by setting 
(t + -1.28125) equal to 3.834592751 and -3.834592751.

Subproblem 1

t + -1.28125 = 3.834592751 Simplifying t + -1.28125 = 3.834592751 Reorder the terms: -1.28125 + t = 3.834592751 Solving -1.28125 + t = 3.834592751 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.28125' to each side of the equation. -1.28125 + 1.28125 + t = 3.834592751 + 1.28125 Combine like terms: -1.28125 + 1.28125 = 0.00000 0.00000 + t = 3.834592751 + 1.28125 t = 3.834592751 + 1.28125 Combine like terms: 3.834592751 + 1.28125 = 5.115842751 t = 5.115842751 Simplifying t = 5.115842751

Subproblem 2

t + -1.28125 = -3.834592751 Simplifying t + -1.28125 = -3.834592751 Reorder the terms: -1.28125 + t = -3.834592751 Solving -1.28125 + t = -3.834592751 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '1.28125' to each side of the equation. -1.28125 + 1.28125 + t = -3.834592751 + 1.28125 Combine like terms: -1.28125 + 1.28125 = 0.00000 0.00000 + t = -3.834592751 + 1.28125 t = -3.834592751 + 1.28125 Combine like terms: -3.834592751 + 1.28125 = -2.553342751 t = -2.553342751 Simplifying t = -2.553342751

Solution

The solution to the problem is based on the solutions from the subproblems. t = {5.115842751, -2.553342751}

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