32.5=65*t-16t^2

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Solution for 32.5=65*t-16t^2 equation:


Simplifying
32.5 = 65t + -16t2

Solving
32.5 = 65t + -16t2

Solving for variable 't'.

Reorder the terms:
32.5 + -65t + 16t2 = 65t + -65t + -16t2 + 16t2

Combine like terms: 65t + -65t = 0
32.5 + -65t + 16t2 = 0 + -16t2 + 16t2
32.5 + -65t + 16t2 = -16t2 + 16t2

Combine like terms: -16t2 + 16t2 = 0
32.5 + -65t + 16t2 = 0

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

Divide each side by '16'.
2.03125 + -4.0625t + t2 = 0

Move the constant term to the right:

Add '-2.03125' to each side of the equation.
2.03125 + -4.0625t + -2.03125 + t2 = 0 + -2.03125

Reorder the terms:
2.03125 + -2.03125 + -4.0625t + t2 = 0 + -2.03125

Combine like terms: 2.03125 + -2.03125 = 0.00000
0.00000 + -4.0625t + t2 = 0 + -2.03125
-4.0625t + t2 = 0 + -2.03125

Combine like terms: 0 + -2.03125 = -2.03125
-4.0625t + t2 = -2.03125

The t term is -4.0625t.  Take half its coefficient (-2.03125).
Square it (4.125976563) and add it to both sides.

Add '4.125976563' to each side of the equation.
-4.0625t + 4.125976563 + t2 = -2.03125 + 4.125976563

Reorder the terms:
4.125976563 + -4.0625t + t2 = -2.03125 + 4.125976563

Combine like terms: -2.03125 + 4.125976563 = 2.094726563
4.125976563 + -4.0625t + t2 = 2.094726563

Factor a perfect square on the left side:
(t + -2.03125)(t + -2.03125) = 2.094726563

Calculate the square root of the right side: 1.447317022

Break this problem into two subproblems by setting 
(t + -2.03125) equal to 1.447317022 and -1.447317022.

Subproblem 1

t + -2.03125 = 1.447317022 Simplifying t + -2.03125 = 1.447317022 Reorder the terms: -2.03125 + t = 1.447317022 Solving -2.03125 + t = 1.447317022 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.03125' to each side of the equation. -2.03125 + 2.03125 + t = 1.447317022 + 2.03125 Combine like terms: -2.03125 + 2.03125 = 0.00000 0.00000 + t = 1.447317022 + 2.03125 t = 1.447317022 + 2.03125 Combine like terms: 1.447317022 + 2.03125 = 3.478567022 t = 3.478567022 Simplifying t = 3.478567022

Subproblem 2

t + -2.03125 = -1.447317022 Simplifying t + -2.03125 = -1.447317022 Reorder the terms: -2.03125 + t = -1.447317022 Solving -2.03125 + t = -1.447317022 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '2.03125' to each side of the equation. -2.03125 + 2.03125 + t = -1.447317022 + 2.03125 Combine like terms: -2.03125 + 2.03125 = 0.00000 0.00000 + t = -1.447317022 + 2.03125 t = -1.447317022 + 2.03125 Combine like terms: -1.447317022 + 2.03125 = 0.583932978 t = 0.583932978 Simplifying t = 0.583932978

Solution

The solution to the problem is based on the solutions from the subproblems. t = {3.478567022, 0.583932978}

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