0=-16t^2+65t+5

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Solution for 0=-16t^2+65t+5 equation:


Simplifying
0 = -16t2 + 65t + 5

Reorder the terms:
0 = 5 + 65t + -16t2

Solving
0 = 5 + 65t + -16t2

Solving for variable 't'.

Combine like terms: 0 + -5 = -5
-5 + -65t + 16t2 = 5 + 65t + -16t2 + -5 + -65t + 16t2

Reorder the terms:
-5 + -65t + 16t2 = 5 + -5 + 65t + -65t + -16t2 + 16t2

Combine like terms: 5 + -5 = 0
-5 + -65t + 16t2 = 0 + 65t + -65t + -16t2 + 16t2
-5 + -65t + 16t2 = 65t + -65t + -16t2 + 16t2

Combine like terms: 65t + -65t = 0
-5 + -65t + 16t2 = 0 + -16t2 + 16t2
-5 + -65t + 16t2 = -16t2 + 16t2

Combine like terms: -16t2 + 16t2 = 0
-5 + -65t + 16t2 = 0

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

Divide each side by '16'.
-0.3125 + -4.0625t + t2 = 0

Move the constant term to the right:

Add '0.3125' to each side of the equation.
-0.3125 + -4.0625t + 0.3125 + t2 = 0 + 0.3125

Reorder the terms:
-0.3125 + 0.3125 + -4.0625t + t2 = 0 + 0.3125

Combine like terms: -0.3125 + 0.3125 = 0.0000
0.0000 + -4.0625t + t2 = 0 + 0.3125
-4.0625t + t2 = 0 + 0.3125

Combine like terms: 0 + 0.3125 = 0.3125
-4.0625t + t2 = 0.3125

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 = 0.3125 + 4.125976563

Reorder the terms:
4.125976563 + -4.0625t + t2 = 0.3125 + 4.125976563

Combine like terms: 0.3125 + 4.125976563 = 4.438476563
4.125976563 + -4.0625t + t2 = 4.438476563

Factor a perfect square on the left side:
(t + -2.03125)(t + -2.03125) = 4.438476563

Calculate the square root of the right side: 2.106769224

Break this problem into two subproblems by setting 
(t + -2.03125) equal to 2.106769224 and -2.106769224.

Subproblem 1

t + -2.03125 = 2.106769224 Simplifying t + -2.03125 = 2.106769224 Reorder the terms: -2.03125 + t = 2.106769224 Solving -2.03125 + t = 2.106769224 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 = 2.106769224 + 2.03125 Combine like terms: -2.03125 + 2.03125 = 0.00000 0.00000 + t = 2.106769224 + 2.03125 t = 2.106769224 + 2.03125 Combine like terms: 2.106769224 + 2.03125 = 4.138019224 t = 4.138019224 Simplifying t = 4.138019224

Subproblem 2

t + -2.03125 = -2.106769224 Simplifying t + -2.03125 = -2.106769224 Reorder the terms: -2.03125 + t = -2.106769224 Solving -2.03125 + t = -2.106769224 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 = -2.106769224 + 2.03125 Combine like terms: -2.03125 + 2.03125 = 0.00000 0.00000 + t = -2.106769224 + 2.03125 t = -2.106769224 + 2.03125 Combine like terms: -2.106769224 + 2.03125 = -0.075519224 t = -0.075519224 Simplifying t = -0.075519224

Solution

The solution to the problem is based on the solutions from the subproblems. t = {4.138019224, -0.075519224}

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