0=16t^2+65t-9

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Solution for 0=16t^2+65t-9 equation:


Simplifying
0 = 16t2 + 65t + -9

Reorder the terms:
0 = -9 + 65t + 16t2

Solving
0 = -9 + 65t + 16t2

Solving for variable 't'.

Combine like terms: 0 + 9 = 9
9 + -65t + -16t2 = -9 + 65t + 16t2 + 9 + -65t + -16t2

Reorder the terms:
9 + -65t + -16t2 = -9 + 9 + 65t + -65t + 16t2 + -16t2

Combine like terms: -9 + 9 = 0
9 + -65t + -16t2 = 0 + 65t + -65t + 16t2 + -16t2
9 + -65t + -16t2 = 65t + -65t + 16t2 + -16t2

Combine like terms: 65t + -65t = 0
9 + -65t + -16t2 = 0 + 16t2 + -16t2
9 + -65t + -16t2 = 16t2 + -16t2

Combine like terms: 16t2 + -16t2 = 0
9 + -65t + -16t2 = 0

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

Divide each side by '-16'.
-0.5625 + 4.0625t + t2 = 0

Move the constant term to the right:

Add '0.5625' to each side of the equation.
-0.5625 + 4.0625t + 0.5625 + t2 = 0 + 0.5625

Reorder the terms:
-0.5625 + 0.5625 + 4.0625t + t2 = 0 + 0.5625

Combine like terms: -0.5625 + 0.5625 = 0.0000
0.0000 + 4.0625t + t2 = 0 + 0.5625
4.0625t + t2 = 0 + 0.5625

Combine like terms: 0 + 0.5625 = 0.5625
4.0625t + t2 = 0.5625

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 = 0.5625 + 4.125976563

Reorder the terms:
4.125976563 + 4.0625t + t2 = 0.5625 + 4.125976563

Combine like terms: 0.5625 + 4.125976563 = 4.688476563
4.125976563 + 4.0625t + t2 = 4.688476563

Factor a perfect square on the left side:
(t + 2.03125)(t + 2.03125) = 4.688476563

Calculate the square root of the right side: 2.165289025

Break this problem into two subproblems by setting 
(t + 2.03125) equal to 2.165289025 and -2.165289025.

Subproblem 1

t + 2.03125 = 2.165289025 Simplifying t + 2.03125 = 2.165289025 Reorder the terms: 2.03125 + t = 2.165289025 Solving 2.03125 + t = 2.165289025 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 = 2.165289025 + -2.03125 Combine like terms: 2.03125 + -2.03125 = 0.00000 0.00000 + t = 2.165289025 + -2.03125 t = 2.165289025 + -2.03125 Combine like terms: 2.165289025 + -2.03125 = 0.134039025 t = 0.134039025 Simplifying t = 0.134039025

Subproblem 2

t + 2.03125 = -2.165289025 Simplifying t + 2.03125 = -2.165289025 Reorder the terms: 2.03125 + t = -2.165289025 Solving 2.03125 + t = -2.165289025 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 = -2.165289025 + -2.03125 Combine like terms: 2.03125 + -2.03125 = 0.00000 0.00000 + t = -2.165289025 + -2.03125 t = -2.165289025 + -2.03125 Combine like terms: -2.165289025 + -2.03125 = -4.196539025 t = -4.196539025 Simplifying t = -4.196539025

Solution

The solution to the problem is based on the solutions from the subproblems. t = {0.134039025, -4.196539025}

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